Serge Cantat

Romain Dujardin

RANDOM DYNAMICS ON REAL AND COMPLEX PROJECTIVE SURFACES

We initiate the study of random iteration of automorphisms of real and complex projective surfaces, as well as compact Kähler surfaces, focusing on the classification of stationary measures. We show that, in a number of cases, such stationary measures are invariant, and provide criteria for uniqueness, smoothness and rigidity of invariant probability measures. This involves a variety of tools from complex and algebraic geometry, random products of matrices, non-uniform hyperbolicity, as well as recent results of Brown and Rodriguez Hertz on random iteration of surface diffeomorphisms.

1. INTRODUCTION 1.1. Random dynamical systems. Consider a compact manifold M and a probability measure ν on DiffpM q; to simplify the exposition we assume throughout this introduction that the support Supppνq is finite. The data pM, νq defines a random dynamical system, obtained by randomly composing independent diffeomorphisms with distribution ν. In this paper, these random dynamical systems are studied from the point of view of ergodic theory, that is, we are mostly interested in understanding the asymptotic distribution of orbits.

A probability measure µ on M is ν-invariant if f ˚µ " µ for ν-almost every f P DiffpM q, and it is ν-stationary if it is invariant on average: ş f ˚µ dνpf q " µ. A simple fixed point argument shows that stationary measures always exist. On the other hand, the existence of an invariant measure should hold only under special circumstances, for instance when the group Γ ν generated by Supppνq is amenable, or has a finite orbit, or preserves an invariant volume form.

According to Breiman's law of large numbers, for every x P M and ν N -almost every pf j q P DiffpM q N , every cluster value of the sequence of empirical measures (1.1)

1 n n´1 ÿ j"0 δ f j ˝¨¨¨˝f 0 pxq
is a stationary measure. Thus, a classification of stationary measures gives an essentially complete understanding of the asymptotic distribution of such random orbits, as n goes to `8. Our goal is to combine algebraic and holomorphic dynamics with recent results in random dynamics to study the case when M is a real or complex projective surface and the action is by algebraic diffeomorphisms. But let us first highlight a nice example to which our techniques apply; rooted in elementary euclidean geometry, it leads to interesting algebro-geometric constructions (see Remarks 3.5 and 3.10).

1.2. Randomly folding pentagons. Let 0 , . . . , 4 be five positive real numbers such that there exists a pentagon with side lengths i . Here a pentagon is just an ordered set of points pa i q i"0,...,4 in the Euclidean plane, such that distpa i , a i`1 q " i for i " 0, . . . , 4 (with a 5 " a 0 by definition); pentagons are not assumed to be convex, and two distincts sides ra i , a i`1 s and ra j , a j`1 s may intersect at a point which is not one of the a i 's. Let Pentp 0 , . . . , 4 q be the set of pentagons with side lengths i . This set may be defined by polynomial equations of the form distpa i , a i`1 q 2 " 2 i , so it is naturally a real algebraic variety. For every i, a i is one of the two intersection points ta i , a 1 i u of the circles of respective centers a i´1 and a i`1 and radii i´1 and i . The transformation exchanging these two points a i and a 1 i , while keeping the other vertices fixed, defines an involution s i of Pentp 0 , . . . , 4 q. It commutes with the action of positive isometries of the plane, hence, it induces an involution σ i on the quotient space (1.2)

Pent 0 p 0 , . . . , 4 q " Pentp 0 , . . . , 4 q{pSO 2 pRq ˙R2 q.

Each element of Pent 0 p 0 , . . . , 4 q admits a unique representative with a 0 " p0, 0q and a 1 " p 0 , 0q, so as before Pent 0 p 0 , . . . , 4 q is a real algebraic variety, which is easily seen to be of dimension 2 (see [START_REF] Curtis | Configuration spaces of planar pentagons[END_REF][START_REF] Shimamoto | Spaces of polygons in the plane and Morse theory[END_REF]). When smooth (see Lemma (3.6) and the comments preceding Remark 3.9 for a discussion), Pent 0 p 0 , . . . , 4 q is a real K3 surface on which the σ i act by algebraic diffeomorphisms, preserving a canonically defined area form (see §3.2); and for a general choice of lengths, the group generated by these involutions generates a rich dynamics. Now, start with some pentagon and at every unit of time, apply randomly one of the σ i . This creates a random sequence of pentagons, and our results explain how this sequence is asymptotically distributed on Pent 0 p 0 , . . . , 4 q. (The dynamics of the folding maps acting on plane quadrilaterals was studied for instance in [START_REF] Esch | The screensaver map: dynamics on elliptic curves arising from polygonal folding[END_REF][START_REF] Benoist | Itération de pliages de quadrilatères[END_REF].)

1.3. Stiffness. Let us present a few landmark results about stationary measures. First, suppose that ν is a finitely supported probability measure on SL 2 pCq, which we view as acting by linear projective transformations on M " P 1 pCq. Suppose that the group Γ ν generated by the support of ν is non-elementary, that is, Γ ν is unbounded and acts strongly irreducibly on C 2 . Then, there is a unique ν-stationary (probability) measure µ on P 1 pCq, and this measure is not invariant. This is one instance of a more general result due to Furstenberg [START_REF] Furstenberg | Noncommuting random products[END_REF]. Now, let ν be a finitely supported measure on SL 2 pZq, and consider the action of SL 2 pZq on the torus M " R 2 {Z 2 . In that case, the Haar measure of R 2 {Z 2 , as well as the atomic measures equidistributed on finite orbits Γ ν px, yq, for px, yq P Q 2 {Z 2 , are examples of Γ ν -invariant measures. By using Fourier analysis and additive combinatorics techniques, Bourgain, Furman, Lindenstrauss and Mozes [START_REF] Bourgain | Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus[END_REF] proved that if Γ ν is non-elementary, then every stationary measure µ on R 2 {Z 2 is Γ ν -invariant and is a convex combination of the above mentioned invariant measures. This property of automatic invariance of stationary measures was called stiffness (or ν-stiffness) by Furstenberg [START_REF] Furstenberg | Stiffness of group actions[END_REF], who conjectured it to hold in this setting. Soon after, Benoist and Quint [START_REF] Benoist | Mesures stationnaires et fermés invariants des espaces homogènes[END_REF] gave an ergodic theoretic proof of this result and extended it to certain actions of discrete groups on homogeneous spaces. They also derived the following equidistribution result: for every px, yq R Q 2 {Z 2 , the random trajectories of px, yq determined by ν almost surely equidistribute towards the Haar measure on R 2 {Z 2 .

Finally, Brown and Rodriguez-Hertz [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF], building on the work of Eskin and Mirzakhani [START_REF] Eskin | Invariant and stationary measures for the SLp2, Rq action on moduli space[END_REF], managed to recast these measure rigidity results in terms of Pesin theory to obtain a version of the stiffness theorem of [START_REF] Bourgain | Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus[END_REF] for general C 2 diffeomorphisms of compact surfaces. We shall describe their results in due time; for the moment we content ourselves with one illustrative consequence of [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF]. Let ν " ř α j δ f j be a finitely supported probability measure on SL 2 pZq generating a non-elementary subgroup. Consider perturbations tf i,ε u of the f i in the group Diff 2 vol pR 2 {Z 2 q of C 2 diffeomorphisms of R 2 {Z 2 preserving the Haar measure. Set ν ε " ř α j δ f j,ε . Then, for sufficiently small perturbations, any ν ε -stationary measure on R 2 {Z 2 is invariant and is a combination of the Haar measure and measures supported on finite Γ νε -orbits.

In this paper, we prove a stiffness theorem for groups of algebraic diffeomorphisms of real algebraic surfaces. The work of Brown and Rodriguez-Hertz is our main source of inspiration and a key ingredient for some of our main results. 1.4. Sample results: stiffness, classification, and rigidity. Let X be a smooth complex projective surface, or more generally a compact Kähler surface. Denote by AutpXq its group of holomorphic diffeomorphisms, referred to in this paper as automorphisms. When X Ă P N pCq is defined by polynomial equations with real coefficients, the complex conjugation induces an anti-holomorphic involution s : X Ñ X, whose fixed point set is the real part XpRq of X. We denote by X R the surface X viewed as an algebraic variety defined over R, and by AutpX R q the group of automorphisms defined over R; AutpX R q is the subgroup of AutpXq centralizing s. When XpRq ‰ H, the elements of AutpX R q are the real-analytic diffeomorphisms of XpRq admitting a holomorphic extension to X. Note that in stark contrast with groups of smooth diffeomorphisms, the groups AutpX R q and AutpXq are typically discrete and at most countable.

The group AutpXq acts on the cohomology H ˚pX ; Zq. By definition, a subgroup Γ Ă AutpXq is non-elementary if its image Γ ˚Ă GLpH ˚pX ; Cqq contains a non-Abelian free group; equivalently, Γ ˚is not virtually Abelian. When Γ is non-elementary, there exists a pair pf, gq P Γ 2 generating a free group of rank 2 such that the topological entropy of every element in that group is positive (see Lemma A.1). Pentagon foldings provide examples for which AutpX R q is non-elementary.

1.4.1. Stiffness. As before, if ν is a finitely supported probability measure on AutpXq, we denote by Γ ν the subgroup generated by Supppνq.

Theorem A. Let X R be a real projective surface and ν be a finitely supported symmetric probability measure on AutpX R q. If Γ ν preserves an area form on XpRq, then every ergodic νstationary measure µ on XpRq is either invariant or supported on a proper Γ ν -invariant subvariety. In particular if there is no Γ ν -invariant algebraic curve, the random dynamical system pX, νq is stiff.

This theorem is mostly interesting when Γ ν is non-elementary and we focus on this case in the remainder of this introduction. Stationary measures supported on invariant curves are easily analysed (see §10.4). Moreover, if Γ ν is non-elementary, it is always possible to contract all Γ νinvariant curves, creating a complex analytic surface X 0 with finitely many singularities. Then on X 0 pRq, stiffness holds unconditionally.

This result applies to many interesting examples, because Abelian, K3, and Enriques surfaces, which concentrate most of the dynamically interesting automorphisms on compact complex surfaces, admit a canonical AutpXq-invariant 2-form. In particular, it applies to pentagon foldings. Note also that linear Anosov maps on R 2 {Z 2 fall into this category, so Theorem A contains the stiffness statement of [START_REF] Bourgain | Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus[END_REF] in the two-dimensional case.

1.4.2. Invariant measures. Once stiffness is established, the next step is to classify invariant measures. A parabolic automorphism of a compact Kähler surface is an automorphism g such that the norm of pg n q ˚on H 2 pX; Rq grows quadratically (as αn 2 for some α ą 0); such an automorphism preserves automatically a genus 1 fibration on X (see e.g. [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF]). An example is given by the composition of the foldings σ i and σ i`1 of two adjacent vertices in the space of pentagons. When Γ ν contains a parabolic automorphism, Γ ν -invariant measures are classified in [START_REF] Cantat | Sur la dynamique du groupe d'automorphismes des surfaces K3[END_REF][START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF]. A nice consequence is that for a non-elementary group of AutpX R q containing parabolic elements and preserving an area form, any invariant ergodic measure is either atomic, or concentrated on a Γ ν -invariant algebraic curve, or is the restriction of the area form on some open subset of XpRq bounded by a piecewise smooth curve.

For random pentagon foldings, these results give a complete answer to the equidistribution problem raised in §1.1. Indeed, assume for simplicity that the group generated by the five involutions σ i of Pent 0 p 0 , . . . , 4 q does not preserve any proper Zariski closed set, and that Pent 0 p 0 , . . . , 4 q is connected. Then the stiffness and classification theorems imply that the only stationary measure is the canonical area form. Therefore by Breiman's law of large numbers, for every initial pentagon P P Pent 0 p 0 , . . . , 4 q and almost every sequence pm j q P t0, . . . , 4u N , the random sequence P n " pσ m n´1 ˝¨¨¨˝σ m 0 qpP q equidistributes with respect to the area form. Thus, quantities like the asymptotic average of the diameter are given by explicit integrals of semi-algebraic functions, independently of the starting pentagon P .

Another widely studied example is the family of Wehler surfaces. These are the smooth surfaces X Ă P 1 ˆP1 ˆP1 defined by an equation of degree p2, 2, 2q. Then for each index i P t1, 2, 3u, the projection π i : X Ñ P 1 ˆP1 which "forgets the variable x i " has degree 2; thus, there is an involution σ i of X that permutes the two points in the generic fiber of π i .

Corollary. Let X R Ă P 1 ˆP1 ˆP1 be a real Wehler surface such that XpRq is non empty. If X R is generic, then:

(1) the surface X is a K3 surface and there is a unique (up to choosing an orientation of XpRq) algebraic 2-form vol X R on XpRq such that ş XpRq vol X R " 1;

(2) the group AutpX R q is generated by the three involutions σ i and coincides with AutpXq;

furthermore it preserves the probability measure defined by vol X R ; (3) if ν is finitely supported and Γ ν has finite index in AutpX R q then pXpRq, νq is stiff; moreover the only ν-stationary measures on XpRq are convex combinations of the probability measures defined by vol X R on the connected components of XpRq.

Here by generic we mean that the equation of X belongs to the complement of at most countably many hypersurfaces in the set of polynomial equations of degree p2, 2, 2q (see §3.1 for details). This result follows from Theorem A, Proposition 3.3, Corollary B of [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF], and the generic non-existence of finite orbits established in [START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF]. If we do not assume X to be generic but assume only that X does not contain any fiber of the three projections π i , then the set of stationary measures supported in XpRq is a finite dimensional simplex (see [START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF]).

The techniques of [START_REF] Cantat | Sur la dynamique du groupe d'automorphismes des surfaces K3[END_REF][START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF] do not apply in the absence of parabolic automorphisms. Here, we establish the following rigidity result.

Theorem B. Let X R be a real projective surface. Let Γ be a non-elementary subgroup of AutpX R q. If µ is a Γ-invariant probability measure on XpRq and if µ is ergodic and of positive entropy for some f P Γ, then µ is absolutely continuous with respect to any area form on XpRq.

In particular if Γ is a group of area preserving automorphisms, then up to normalization µ will be the restriction of the area form on some Γ-invariant set. Kummer examples are a generalization of linear Anosov diffeomorphisms of tori to other projective surfaces (see [START_REF] Cantat | Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy[END_REF][START_REF] Cantat | Holomorphic actions, Kummer examples, and Zimmer program[END_REF]). When Γ contains a real Kummer example, we can derive an exact analogue of the classification of invariant measures of [START_REF] Bourgain | Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus[END_REF], that is the assumption "µ has positive entropy" can be replaced by "µ has no atoms" (Theorem 11.4). We also obtain a version of Theorem B for polynomial automorphisms of the affine plane A 2 R (see Theorem 11.5).

1.5. Some ingredients of the proofs. The proofs of Theorems A and B rely on the deep results of Brown and Rodriguez-Hertz [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF]. To be more precise, recall that an ergodic stationary measure µ on X admits a pair of Lyapunov exponents λ `pµq ě λ ´pµq, and that µ is called hyperbolic if λ `pµq ą 0 ą λ ´pµq. In this case the (random) Oseledets theorem shows that for µ-almost every x and ν N -almost every ω " pf j q jPN in AutpXq N , there exists a stable direction E s ω pxq Ă T x X R . In [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF], stiffness is established under the condition that E s ω pxq Ă T x X R depends non-trivially on the random itinerary ω " pf j q jPN , or equivalently that stable directions do not induce a measurable Γ ν -invariant line field. One of our main contributions is to take care of this possibility in our setting: for this we study the dynamics on the complex surface X.

Theorem C. Let X be a complex projective surface and ν be a finitely supported probability measure on AutpXq. If Γ ν is non-elementary, then any hyperbolic ergodic ν-stationary measure µ on X satisfies the following alternative:

(a) either µ is invariant, and its fiber entropy h µ pX; νq vanishes; (b) or µ is supported on a Γ ν -invariant algebraic curve; (c) or the field of Oseledets stable directions of µ is not Γ ν -invariant; in other words, it genuinely depends on the itinerary ω " pf j q jě0 P AutpXq N .

As opposed to Theorems A and B, this result applies to the dynamics on the complex manifold X, without assuming the existence of an invariant volume form or an invariant real structure. When µ is not invariant, nor supported by a proper Zariski closed subset, Assertion (c) precisely says that the condition on stable directions used in [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF] is satisfied. This is our key input towards Theorems A and B. The arguments leading to Theorem C involve an interesting blend of Hodge theory, pluripotential analysis, and Pesin theory. They rely on the following well-known principle in higher dimensional holomorphic dynamics. If µ is ergodic and hyperbolic, almost every point pω, xq provides a stable manifold W s ω pxq biholomorphic to C. Then, according to a construction going back to Ahlfors and Nevanlinna, to any entire curve φ : C Ñ X is associated a (family of) closed positive p1, 1q-current(s) describing the asymptotic distribution of φpCq in X, hence also a (family of) cohomology class(es) in H 2 pX, Rq. These classes link the stable manifolds of µ with the action of Γ ν on H 2 pX; Rq, which itself can be analyzed by combining tools from complex algebraic geometry with Furstenberg's theory of random products of matrices.

Theorem D. Let X be a complex projective surface. Let ν be a finitely supported probability measure on AutpXq such that Γ ν is non-elementary. Let κ 0 be a fixed Kähler form on X.

(1) If κ is any Kähler form on X, then for ν N -almost every ω :" pf j q jě0 P AutpXq N the limit T s ω :" lim nÑ`8 1 ş X κ 0 ^pf n ˝¨¨¨˝f 0 q ˚κ pf n ˝¨¨¨˝f 0 q ˚κ exists as a closed positive p1, 1q-current. Moreover this current T s ω does not depend on κ and has Hölder continuous potentials.

(2) If the ν-stationary measure µ is ergodic, hyperbolic (or more generally if λ ´pµq ă 0 ď λ `pµq) and not supported on a Γ ν -invariant proper Zariski closed set, then for µ-almost every x and ν N -almost every ω, the only Ahlfors-Nevanlinna current of mass 1 (with respect to κ 0 ) associated to the stable manifold W s ω pxq coincides with T s ω .

The right setting for such a statement is that of a compact Kähler surface, but we show in §3.6 that any compact Kähler surface supporting a non-elementary group of automorphisms is projective (see Appendix A for the non-Kähler case). The algebraicity of X is, in fact, a crucial technical ingredient in the proof of assertion [START_REF] Bahnmüller | A Margulis-Ruelle inequality for random dynamical systems[END_REF], because we use techniques of laminar currents which are available only on projective surfaces. Theorem D enters the proof of Theorem C as follows: since Γ ν is non-elementary, Furstenberg's description of the random action on H 2 pX, Rq implies that the cohomology class rT s ω s depends non-trivially on ω; therefore for µ-almost every x, W s ω pxq also depends non-trivially on ω. Then, taking advantage of the complex structure again, we show in Section 9, that E s ω pxq depends non-trivially on ω as well.

Remark 1.1. Beyond finitely supported measures, Theorem A, B, C, and D hold under optimal moment conditions on ν (this adds several technicalities, notably in Sections 5 and 6).

1.6. Organization of the article. Let X be a compact Kähler surface and ν be a probability measure on AutpXq.

-In Section 2 we describe the action of AutpXq on H ˚pX ; Zq, in particular on H 1,1 pX; Rq. The Hodge index theorem endows it with a Minkowski structure, which is essential in our understanding of the dynamics of Γ ν on the cohomology. This section 2 prepares the ground for the analysis of random products of matrices done in Section 5 (and it is also used in [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF][START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF]). A delicate point to keep in mind is that the action of a non-elementary subgroup of AutpXq on H 1,1 pX; Rq may be reducible. -Section 3 describes several classes of examples, including pentagon foldings and Wehler's surfaces. It is also shown there that a compact Kähler surface with a non-elementary group of automorphims is necessarily projective (see Theorem E in §3. [START_REF] Barth | ume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]). -After a short Section 4 introducting the vocabulary of random products of diffeomorphisms, Furstenberg's theory of random products of matrices is applied in Section 5 to the study of the action on H 1,1 pX; Rq. This, combined with the theory of closed positive currents, leads to the proof of the first assertion of Theorem D in Section 6 (see Corollary 6.12 and Theorems 6.14 and 6. [START_REF] Bost | Heights of projective varieties and positive Green forms[END_REF]). The continuity of the potentials of the currents T s ω , which plays a key role in Section 8, relies on a recent result of Gouëzel and Karlsson [START_REF] Gouëzel | Subadditive and multiplicative ergodic theorems[END_REF].

-Pesin theory enters into play in Section 7, in which the basics of the smooth ergodic theory of random dynamical systems are described in some detail for complex surfaces. This is used in Section 8 to connect the stable manifolds to the currents T s ω , using techniques of laminar currents (Theorem 8.2 gives the second part of Theorem D).

-Theorem C is proven in Section 9 by combining ideas of [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF] with Theorem D and an elementary fact from local complex geometry inspired by a lemma from [7]. -Theorem A is finally established in Section 10. When Γ ν is non-elementary (Theorem 10.10) it follows rather directly from [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF], Theorem C, and the invariance principle of Crauel [START_REF] Crauel | Non-Markovian invariant measures are hyperbolic[END_REF].

Elementary groups are handled separately by using the classification of automorphism groups of compact Kähler surfaces (see Theorems 10.3 and Proposition 10.5). Note that the symmetry of ν is used only in the elementary case. -Section 11, as well as [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF], are devoted to the classification of invariant measures. We prove Theorem B in the more precise form of Theorem 11.1, as well as several related results.

1.7. Further comments.

-This article is part of a series of papers dedicated to the dynamics of groups of automorphisms of compact Kähler surfaces, notably K3 and Enriques surfaces. The article [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF] focuses on the classification of invariant measures in presence of parabolic elements, and in [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces: hyperbolicity[END_REF] we study the hyperbolicity of such measures. In [START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF] the existence of finite orbits is analyzed by using tools from algebraic and arithmetic dynamics. -After the first version of this paper and [START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF] were released, Filip and Tosatti [START_REF] Filip | Canonical currents and heights for k3 surfaces[END_REF] gave an alternate approach of some of the results of Section 6. -It is natural to wonder what remains of our results in the real-analytic category. As explained above, the proofs of Theorems D and C rely on a global complex geometric argument (via Ahlfors-Nevanlinna currents and cohomology groups) to show that stable manifolds depend on random itineraries, and on local properties of complex analytic disks to go from stable manifolds to stable directions (cf. Section 9). It is clear that the global arguments do not carry over to groups of real-analytic diffeomorphisms of closed real surfaces. To be more explicit, consider the following fact: if f is an automorphism of a complex projective surface with positive entropy, and if W s f pxq and W u f px 1 q are Zariski dense stable and unstable manifolds of saddle periodic points, then W s f pxq X W u f px 1 q is non-empty. This can be derived from the same global strategy, namely Ahlfors-Nevanlinna currents, their laminarity, and the Hodge index theorem (see [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF]Thm. 6.2]). On the other hand, such a statement does not hold for real analytic diffeomorphisms of closed surfaces. Regarding the results of Section 9, while some of the results of § 9.2 might persist in the real-analytic category, the key Lemma 9.7 does not (see Remark 9.8). -While not directly covered by this article, the character variety of the once punctured torus (or the four times punctured sphere) should be amenable to the same strategy (see [START_REF] Cantat | Painlevé and Schrödinger[END_REF][START_REF] William | Topological components of spaces of representations[END_REF][START_REF] William | The modular group action on real SLp2q-characters of a one-holed torus[END_REF], [START_REF] Previte | Topological dynamics on moduli spaces[END_REF][START_REF] Previte | Topological dynamics on moduli spaces[END_REF], and [START_REF] Liu | A large deviation theorem for random walks on the surface (chapter of liu's ph[END_REF][START_REF] Ngai | Stationary measures and orbit closures of uniformly expanding random dynamical systems on surfaces[END_REF] for a short selection of papers on the subject). -In a forthcoming work, we intend to extend the results of Brown and Rodriguez-Hertz to the complex setting; with Theorem C at hand, this would extend Theorem A from the real to the complex case.

1.8. Conventions. Throughout the paper C stands for a "constant" which may change from line to line, independently of some asymptotic quantity that should be clear from the context (typically an integer n corresponding to the number of iterations of a dynamical system). We write a À b if a ď Cb and ab if a À b À a. Complex manifolds are considered to be connected, so from now on "complex manifold" stands for "connected complex manifold". For a random dynamical system on a disconnected complex manifold, there is a finite index sugbroup Γ 1 of Γ ν fixing each connected component, and an induced measure ν 1 on Γ 1 with properties qualitatively similar to those of ν (see §10.2), so the problem is reduced to the connected case.

1.9. Acknowledgments. We are grateful to Sébastien Gouëzel, Franc ¸ois Ledrappier, and Franc ¸ois Maucourant for interesting discussions and insightful comments. The first named author was partially supported by a grant from the French Academy of Sciences (Del Duca foundation), and the second named author by a grant from the Institut Universitaire de France.

HODGE INDEX THEOREM AND MINKOWSKI SPACES

In this section we define the notion of a non-elementary group of automorphisms of a compact Kähler surface X. We study the action of such a group on the cohomology of X, and in particular the question of (ir)reducibilty. We refer to Appendix A for a discussion of the non-Kähler case.

H k pX; Cq " à p`q"k
H p,q pX; Cq is AutpXq-invariant. On H 0,0 pX; Cq and H 2,2 pX; Cq, AutpXq acts trivially. Throughout the paper we denote by rαs the cohomology class of a closed differential form (or current) α.

The intersection form on H 2 pX; Zq will be denoted by x¨| ¨y; the self-intersection xa|ay of a class a will also be denoted by a 2 for simplicity. This intersection form is AutpXq-invariant. By the Hodge index theorem, it is positive definite on the real part of H 2,0 pX; Cq ' H 0,2 pX; Cq and it is non-degenerate and of signature p1, h 1,1 pXq ´1q on H 1,1 pX; Rq. Thus, we get: Lemma 2.1. The restriction of AutpXq ˚to the subspace H 2,0 pX; Cq (resp. H 0,2 pX; Cq) is contained in a compact subgroup of GLpH 2,0 pX; Cqq (resp. GLpH 0,2 pX; Cqq).

The Néron-Severi group NSpX; Zq is, by definition, the discrete subgroup of H 1,1 pX; Rq defined by NSpX; Zq " H 1,1 pX; Rq X H 2 pX; Zq; more precisely, it is the intersection of H 1,1 pX; Rq with the image of H 2 pX; Zq in H 2 pX; Rq, i.e. with the torsion free part of the Abelian group H 2 pX; Zq. The Lefschetz theorem on p1, 1q-classes identifies NSpX; Zq with the subgroup of H 1,1 pX; Rq given by Chern classes of line bundles on X. The Néron-Severi group is AutpXq-invariant, as well as NSpX; Rq :" NSpX; If u is an element of H 1,0 pX; Cq, then u ^u is an element of H 1,1 pX; Rq such that |u| 2 ď C |u ^u| for some constant C that depends only on the choice of norm on the cohomology; in particular, the norm of f ˚on H 1,0 pX; Cq is controlled by the norm of f ˚on H 1,1 pX; Cq. Using complex conjugation, the same results hold on H 0,1 pX; Cq; by Poincaré duality we also control }f ˚}H p,q pX;Cq for p `q ą 2. Together with Lemma 2.1, we obtain: Lemma 2.2. Let X be a compact Kähler surface. There exists a constant C 0 ą 1 such that C ´1 0 }f ˚}H ˚pX ;Cq ď }f ˚}H 1,1 pX;Rq ď }f ˚}H ˚pX ;Cq for every automorphism f P AutpXq.

Zq b Z R for R " Q, R, or C.

2.2.

The Kähler, nef, and pseudo-effective cones. (See [START_REF] Boucksom | Divisorial Zariski decompositions on compact complex manifolds[END_REF][START_REF] Lazarsfeld | Positivity in algebraic geometry. I[END_REF] for details on the notions introduced in this section.)

Let KahpXq Ă H The intersection NSpX; RqXKahpXq is the ample cone AmppXq, while NSpX; RqXKahpXq is the nef cone NefpXq. They are all invariant under the action of AutpXq on H 1,1 pX; Rq. We shall also say that the elements of KahpXq are nef classes, but the notation NefpXq will be reserved for NSpX; Rq X KahpXq. The set of classes of closed positive currents is the pseudoeffective cone PsefpXq. This cone is an AutpXq-invariant, salient, closed, convex cone. It is dual to KahpXq for the intersection form (see [START_REF] Boucksom | Divisorial Zariski decompositions on compact complex manifolds[END_REF]Lem. 4.1]):

(2.3) KahpXq " tu P H 1,1 pX; Rq ; xu | vy ě 0 @v P PsefpXqu and vice-versa. We fix once and for all a reference Kähler form κ 0 with rκ 0 s 2 " ş κ 0 ^κ0 " 1. Then we define the mass of a pseudo-effective class a by Mpaq " xa | rκ 0 sy, or equivalently the mass of a closed positive current T by MpT q " ş T ^κ0 ; we may also extend this definition to any class, pseudo-effective or not (but then Mpaq " xa | rκ 0 sy may be negative). The compactness of the set of closed positive currents of mass 1 implies that, for any norm |¨| on H 1,1 pX, Rq, there exists a constant C such that (2.4) @a P PsefpXq,

C ´1 |a| ď Mpaq ď C |a| .
If v is an element of PsefpXq and v 2 ě 0, then by the Hodge index theorem we know that xu | vy ě 0 for every class u P H 1,1 pX; Rq such that u 2 ě 0 and xu | rκ 0 sy ě 0 (see Equation (2.6)). So, in Equation (2.3), the most important constraints come from the classes v P PsefpXq with v 2 ă 0. If v is such a class, its Zariski decomposition expresses v as a sum v " ppvq `npvq with the following properties (see [START_REF] Boucksom | Divisorial Zariski decompositions on compact complex manifolds[END_REF]):

(1) this decomposition is orthogonal: xppvq | npvqy " 0;

(2) ppvq is a nef class, i.e. ppvq P KahpXq;

(3) npvq is negative: it is a sum npvq " ř i a i rD i s with positive coefficients a i P R ˚of classes of irreducible curves D i Ă X such that the Gram matrix pxD i | D j yq is negative definite.

Proposition 2.3. If a ray R `v of the cone PsefpXq is extremal, then either v 2 ě 0 or R `v " R `rDs for some irreducible curve D such that D 2 ă 0. The cone PsefpXq contains at most countably many extremal rays R `v with v 2 ă 0.

Let u be an isotropic element of KahpXq. If R `u is not an extremal ray of PsefpXq, then u is proportional to an integral class u 1 P NSpX; Zq.

Proof. If R `v is extremal, the Zariski decomposition v " ppvq `npvq involves only one term. If v " ppvq then v 2 ě 0. Otherwise v " npvq and by extremality npvq " arDs for some irreducible curve D with D 2 ă 0. The countability assertion follows, because NSpX; Zq is countable. For the last assertion, multiply u by xu|rκ 0 sy ´1 to assume xu|rκ 0 sy " 1 and write u as a convex combination u " ş v dαpvq, where α is a probability measure on PsefpXq such that α-almost every v satisfies -xv|rκ 0 sy " 1, -R `v is extremal in PsefpXq and does not contain u.

Since u is nef, xu | vy ě 0 for each v; and u being isotropic, we get v P u K zRu for α-almost every v. By the Hodge index theorem, v 2 ă 0 almost surely. Now, the first assertion of this proposition implies that v P R `rD v s for some irreducible curve D v Ă X with negative selfintersection; there are only countably many classes of that type, thus α is purely atomic, and u belongs to VectprD v s; αpvq ą 0q, a subspace of NSpX; Rq defined over Q. On this subspace, q X is semi-negative, and by the Hodge index theorem its kernel is Ru. Since VectprD v s; αpvq ą 0q and q X are defined over Q, we deduce that u is proportional to an integral class.

2.3.

Non-elementary subgroups of AutpXq. When X is a compact Kähler surface, the action of AutpXq on H 1,1 pX, Rq is subject to several constraints: the Hodge index theorem implies that it must preserve a Minkowski structure and in addition it preserves the lattice given by the Neron-Severi group. In this section we review the first consequences of these constraints.

2.3.1.

Isometries of Minkowski spaces. Consider the Minkowski space R m`1 , endowed with its quadratic form q of signature p1, mq defined by (2.5) qpxq " x 2 0 ´m ÿ i"1

x 2 i .
The corresponding bilinear form will be denoted x¨|¨y. For future reference, note the following reverse Schwarz inequality:

(2.6) if qpxq ě 0 and qpx 1 q ě 0 then xx | x 1 y ě qpxq 1{2 qpx 1 q 1{2

with equality if and only if x and x 1 are collinear. We say that a subspace W Ă R m`1 is of Minkowski type if the restriction q |W is non-degenerate and of signature p1, dimpW q ´1q.

In this section, we review some well-known facts concerning isometries of R 1,m " pR m`1 , qq (see e.g. [START_REF] Ratcliffe | Foundations of hyperbolic manifolds[END_REF][START_REF] Kapovich | Hyperbolic manifolds and discrete groups[END_REF][START_REF] Franchi | Hyperbolic dynamics and Brownian motion[END_REF] for details). We denote by |¨| the Euclidean norm on R m`1 , and by P : R m`1 zt0u Ñ PpR m`1 q the projection on the projective space PpR m`1 q " P m pRq.

The hyperboloid tx ; qpxq " 1u has two components, and we denote by O 1,m pRq the subgroup of the orthogonal group O 1,m pRq that preserves the component Q " tqpxq " 1 ; x 0 ą 0u. Endowed with the distance d H px, yq " cosh ´1xx | yy, Q is a model of the real hyperbolic space H m of dimension m. The boundary at infinity of H m will be identified with BPpQq Ă PpR m`1 q and will be denoted by BH m . It is the set of isotropic lines of q.

Any isometry γ of H m is induced by an element of O 1,m pRq, and extends continuously to BH m : its action on BH m is given by its linear projective action on PpR m`1 q. Isometries are classified in three types, according to their fixed point set in The group O 1,m pRq admits a Cartan or KAK decomposition. To state it, denote by e 0 " p1, 0, . . . , 0q the first vector of the canonical basis of R m`1 ; this vector is an element of H m , and its stabilizer Stabpe 0 q in O 1,m pRq is a maximal compact subgroup, isomorphic to O m´1 pRq. Lemma 2.4 (See §I.5 of [START_REF] Franchi | Hyperbolic dynamics and Brownian motion[END_REF]). Every γ P O 1,m pRq can be written (non-uniquely) as γ " k 1 ak 2 , where k i P Stabpe 0 q and a is a matrix of the form ¨cosh r sinh r 0 sinh r cosh r 0 0 0 id m´1 ' with r " d H pe 0 , γe 0 q.

H m Y BH m : -γ is elliptic if γ has a fixed point in H m ; -γ is parabolic if γ has no
As an immediate corollary, we get:

Corollary 2.5. If }¨} denotes the operator norm associated to the euclidean norm in R m`1 , then }γ} " }a}, where γ " k 1 ak 2 is any Cartan decomposition of γ. In particular }γ} "

› › γ ´1› › and }γ} -cosh d H pe 0 , γpe 0 qq -|γe 0 | .
Furthermore for every e P H m and any γ P O 1,m pRq }γ} -cosh d H pe, γpeqq, where the implied constant depends only on the base point e.

Irreducibility.

A non-elementary subgroup of O 1,m pRq does not need to act irreducibly on R m`1 . Proposition 2.8, below, clarifies the possible situations. Lemma 2.6. Let Γ be a non-elementary subgroup of O 1,m pRq (resp. γ be an element of O 1,m pRq). Let W be a subspace of R 1,m .

(1) If W is Γ-invariant, then either pW, q| W q is a Minkowski space and Γ| W is non-elementary, or q| W is negative definite and Γ| W is contained in a compact subgroup of GLpW q. (2) If W is γ-invariant and contains a vector w with qpwq ą 0, then γ| W has the same type (elliptic, parabolic, or loxodromic) as γ; in particular, W contains the γ-invariant isotropic lines if γ is parabolic or loxodromic.

Proof. The restriction q| W is either a Minkowski form or is negative definite. Indeed, it cannot be positive definite, because W would then be a Γ-invariant line intersecting the hyperbolic space H m in a fixed point; and it cannot be degenerate, since otherwise its kernel would give a Γ-invariant point on BH m . If q| W is a Minkowski form and Γ| W is elementary, then Γ preserves a finite subset of pH m Y BH m q X W and Γ itself is elementary. This proves the first assertion. The proof of the second one is similar.

Let Γ be a non-elementary subgroup of O G " ZarpΓq irr the neutral component of ZarpΓq, for the Zariski topology. Note that the Lie group GpRq is not necessarily connected for the euclidean topology.

Lemma 2.7 (see [START_REF] Cantat | The geometric Bogomolov conjecture[END_REF], §4.1). The group Γ X GpRq has finite index in Γ. If Γ 0 is a finite index subgroup of Γ, then ZarpΓ 0 q irr " G.

Proposition 2.8. Let Γ Ă O 1,m pRq be non-elementary.

(1) The representation of ΓXGpRq (resp. of GpRq) on R 1,m splits as a direct sum of irreducible representations, with exactly one irreducible factor of Minkowski type:

R 1,m " V `' V 0 ;
here V `is of Minkowski type, and V 0 is an orthogonal sum of irreducible representations V 0,j on which the quadratic form q is negative definite.

(2) The restriction G| V `coincides with SOpV `; q| V `q.

(3) The subspaces V `and V 0 are Γ-invariant, and the representation of Γ on V `is strongly irreducible.

Proof. A group Γ is non-elementary if and only if any of its finite index subgroups is nonelementary. So, we can apply Lemma 2.6 to Γ X GpRq:

if W Ă R 1,m is a non-trivial pΓ X GpRqq-invariant subspace, q| W is non-degenerate. As a consequence, R 1,m is the direct sum W ' W K
, where W K is the orthogonal complement of W with respect to q. This implies that the representation of Γ X GpRq on R 1,m splits as a direct sum of irreducible representations, with exactly one irreducible factor of Minkowski type, as asserted in [START_REF] Avila | Extremal Lyapunov exponents: an invariance principle and applications[END_REF].

The group G preserves this decomposition, and by Proposition 1 of [START_REF] Benoist | Adhérence de Zariski des groupes de Coxeter[END_REF], the restriction G| V coincides with SOpV `; q| V `q; this group is isomorphic to the almost simple group SO 1,k pRq, with 1 `k " dimpV `q. This proves the second assertion.

Since G is normalized by Γ, we see that for any γ P Γ, γV `is a G-invariant subspace of the same dimension as V `and on which q is of Minkowski type. Hence V `, as well as its orthogonal complement V 0 are Γ-invariant. By Lemma 2.7, the action of Γ on V `is strongly irreducible; indeed, if a finite index subgroup Γ 0 in Γ preserves a non-trivial subspace of V ` then, by Zariski density of Γ 0 X GpRq in GpRq, this subspace must be V `itself. On V 0 , Γ permutes the irreducible factors V 0,j . Now, set V " R 1,m and assume that there is a lattice

V Z Ă V such that (i) V Z is Γ-invariant;
(ii) the quadratic form q is an integral quadratic form on V Z .

In other words, there is a basis of V with respect to which q and the elements of Γ are given by matrices with integer coefficients. In particular, V has a natural Q-structure, with V pQq " V Z b Z Q. This situation naturally arises for the action of automorphisms of compact Kähler surfaces on NSpX; Rq. The next lemma will be useful in [START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF].

Lemma 2.9. If Γ contains a parabolic element, the decomposition V `' V 0 is defined over Q, Γ| V 0 is a finite group, and G is the subgroup SOpV `; qq ˆtid V 0 u of OpV ; qq.

Proof. If γ P Γ is parabolic, it fixes pointwise a unique isotropic line, therefore this line is defined over Q. In addition it must be contained in V `because pγ n puqq ně0 converges to the boundary point determined by this line for every u P H m . So, V `contains at least one non-zero element of V Z . Since the action of Γ on V `is irreducible, the orbit of this vector generates V ànd is contained in V Z , so V `is defined over Q. Its orthogonal complement V 0 is also defined over Q, because q itself is defined over Q. As a consequence, Γ| V 0 preserves the lattice V 0 X V Z and the negative definite form q| V 0 ; hence, it is finite. Thus G| V 0 is trivial and the last assertion follows from the above mentioned equality G| V `" SOpV `; q| V `q.

Example 2.10. The purpose of this example is to show that the existence of a parabolic element in Γ is indeed necessary in Lemma 2.9, even for a group of automorphisms of a K3 surface.

Let a be a positive square free integer, for instance a " 7 or 15. Let α be the positive square root ? a, K be the quadratic field Qpαq, and η be the unique non-trivial automorphism of K, sending α to its conjugate α :" ηpαq " ´?a. We view η as a second embedding of K in C. Let O K be the ring of integers of K.

Let be an integer ě 2. Consider the quadratic form in `1 variables defined by (2.8) q px 0 , x 1 , . . . , x q " αx 2 0 ´x2 1 ´¨¨¨´x 2 . It is non-degenerate and its signature is p1, q. The orthogonal group Opq ; O K q is a lattice in the real algebraic group Opq , Rq. The conjugate quadratic form q " αx 2 0 ´x2

1 ´¨¨¨´x 2 is negative definite.

Embed O `1 K into R 2 `2 by the map px i q Þ Ñ px i , ηpx i qq, to get a lattice Λ Ă R 2 `2 and consider the quadratic form Q :" q ' q . Then embed Opq ; O K q into OpQ ; Rq by the homomorphism A P Opq , O K q Þ Ñ A ' ηpAq; we denote its image by Γ ˚ Ă OpQ ; Rq. It is shown in [START_REF] Witte | Introduction to arithmetic groups[END_REF], Chapter 6.4, that -Q is defined over Z with respect to Λ, -Γ ˚ Ă OpQ ; Zq (with respect to this integral structure), -the group G " ZarpΓ ˚ q irr coincides with SOpq ; Rq ˆSO 0 pq ; Rq (and the group ηpOpq ; O K qq is dense in the compact group Opq ; Rq). Now, assume 2 ď ď 4, so that 2 `2 ď 10, and change Q into 4Q : it is an even quadratic form on the lattice Λ » Z 2 `2. According to [98, Corollary 2.9], there is a complex projective K3 surface X for which pNSpX; Zq, q X q is isometric to pΛ, 4Q q. On such a surface, the selfintersection of every curve is divisible by 4 and consequently there is no p´2q-curve. So, by the Torelli theorem for K3 surfaces (see [START_REF] Barth | ume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]), AutpXq |NSpX;Zq has finite index in Op4Q ; Zq.

Since Op4Q ; Zq " OpQ ; Zq we can view Γ ˚ as a subgroup of Op4Q ; Zq. Set Γ ˚"

AutpXq ˚X Γ ˚ and let Γ denote its pre-image in AutpXq. Then, Γ is a subgroup of AutpXq for which the decomposition NSpX; Rq `' NSpX; Rq 0 is non-trivial (here, both have dimension `1) while the representation is irreducible over Q. According to the classification of isometries of hyperbolic spaces, there are three types of automorphisms: elliptic, parabolic and loxodromic. An important fact for us is that the type of isometry is related to the dynamics on X; for instance, every parabolic automorphism preserves a genus 1 fibration, every loxodromic automorphism has positive topological entropy (see [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF] for more details). A subgroup Γ of AutpXq is said to be non-elementary if its action on H X is non-elementary. As we shall see below, the existence of such a subgroup forces X to be projective; for expository reasons, the proof of this result is postponed to §3.6.2, Theorem E. 2.3.4. Automorphisms and Néron-Severi groups. Let X be a compact Kähler surface and Γ be a non-elementary subgroup of AutpXq. Let Γ p,q be the image of Γ in GLpH p,q pX; Cqq, and Γ be its image in GLpH 2 pX; Cqq. If we combine Proposition 2.8 together with Lemma 2.1 for Γ 1,1 , we get an invariant decomposition (2.9) H 1,1 pX; Rq " H This is a Minkowski space on which Γ acts strongly irreducibly; the intersection form is negative definite on the orthogonal complement (2.12) Π K Γ Ă H 1,1 pX; Rq. Moreover by Proposition 2.8.(2) the group G " ZarpΓq irr satisfies GpRq| Π Γ " SOpΠ Γ q. If Γ contains a parabolic element, then Π Γ is rational with respect to the integral structures of NSpX; Zq and H 2 pX; Zq, and GpRq " SOpΠ Γ q ˆtid Π K Γ u (see Lemma 2.9). 2.3.5. Invariant algebraic curves I. Assume that Γ is non-elementary and let C Ă X be an irreducible algebraic curve with a finite Γ-orbit. Then the action of Γ on Vect Z tf ˚rCs; f P Γu Ă NSpX; Zq factors through a finite group. From Propositions 2.8 and 2.11 we deduce that the intersection form is negative definite on Vect Z pΓ ¨rCsq, thus Vect R pΓ ¨rCsq is one of the irreducible factors of NSpX, Rq 0 . This argument, together with Grauert's contraction theorem, leads to the following result (we refer to [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF][START_REF] Kawaguchi | Projective surface automorphisms of positive topological entropy from an arithmetic viewpoint[END_REF] for a proof; the result holds more generally for subgroups containing a loxodromic element): Lemma 2.13. Let X be a compact Kähler surface and Γ be a non-elementary group of automorphisms on X. Then, there are at most finitely many Γ-periodic irreducible curves. The intersection form is negative definite on the subspace of NSpX; Zq generated by the classes of these curves. There is a compact complex analytic surface X 0 and a Γ-equivariant bimeromorphic morphism X Ñ X 0 that contracts these curves and is an isomorphism in their complement.

The next result follows from [START_REF] Diller | Invariant curves for birational surface maps[END_REF].

Proposition 2.14. Let X be a compact Kähler surface and Γ a non-elementary subgroup of AutpXq. Then any Γ-periodic curve has arithmetic genus 0 or 1.

Note if C is Γ-periodic, this result applies to r C " Γ ¨C, which is invariant. Then, the normalization of any irreducible component of r C has genus 0 or 1, and the incidence graph of the components of r C obeys certain restrictions (see [27, §4.1] for details). If furthermore X is a K3 or Enriques surface, each component is a smooth rational curve of self-intersection ´2.

2.3.6. The limit set. Let Γ Ă AutpXq be non-elementary. The limit set of Γ is the closed subset LimpΓq Ă BH X Ă P `H1,1 pX; Rq ˘defined by one of the following equivalent assertions:

(a) LimpΓq is the smallest, non-empty, closed, and Γ-invariant subset of PpH X q; (b) LimpΓq Ă BH X is the closure of the set of fixed points of loxodromic elements of Γ in BH X ; (c) LimpΓq is the accumulation set of any Γ-orbit ΓpPpvqq Ă PpH 1,1 pX; Rqq, for any v R Π K Γ . We refer to [START_REF] Kapovich | Hyperbolic manifolds and discrete groups[END_REF][START_REF] Ratcliffe | Foundations of hyperbolic manifolds[END_REF] for a study of such limit sets. From the second characterization we get: Lemma 2.15. The limit set LimpΓq of a non-elementary group is contained in PpΠ Γ q X BH X .

From the third characterization, LimpΓq is contained in the closure of ΓpPprκsqq for every Kähler form κ on X. Since X must be projective, we can chose rκs in NSpX; Zq. As a consequence, LimpΓq is contained in NefpXq: Lemma 2.16. Let X be a compact Kähler surface. If Γ is a non-elementary subgroup of AutpXq its limit set satisfies LimpΓq Ă PpNefpXqq Ă PpNSpX; Rqq.

2.4. Parabolic automorphisms. We collect a few basic facts on parabolic automorphisms: they will be used in the next section to describe explicit examples, and then in Section 10.

Let f be a parabolic automorphism of a compact Kähler surface. Then f ˚preserves a unique point on BH X , and f preserves a unique genus 1 fibration π f : X Ñ B onto some Riemann surface B. The fixed point of f ˚on BH X is given by the class rF s of any fiber of π f (see [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF]). The fibers of π f are the elements of the linear system |F |, π f is uniquely determined by rF s, and if g is another automorphism of X that preserves a smooth fiber of π f (resp. the point PrF s P PNSpX; Rq), then g preserves the fibration and is either elliptic or parabolic.

Lemma 2.17. Let X be a K3 or Enriques surface, and π : X Ñ B be a genus 1 fibration. If g P AutpXq maps some fiber F of π to a fiber of π, then g preserves the fibration and either g is parabolic or it is periodic of order ď 66.

Proof. Since g maps F to some fiber F 1 , it maps the complete linear system |F | to |F 1 |, but both linear systems are made of the fibers of π. So g preserves the fibration and is not loxodromic. If g is not parabolic it is elliptic, and its action on cohomology has finite order since it preserves H 2 pX, Zq. On a K3 or Enriques surface every holomorphic vector field vanishes identically, so AutpXq 0 is trivial and the kernel of the homomorphism AutpXq Q f Þ Ñ f ˚is finite (see [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF]Theorem 2.6]); as a consequence, any elliptic automorphism has finite order. The upper bound on the order of g was obtained in [START_REF] Keum | Orders of automorphisms of K3 surfaces[END_REF].

Proposition 2.18. Let X be a compact Kähler surface and let f be a parabolic automorphism of X, preserving the genus 1 fibration τ : X Ñ B. Consider the group AutpX; τ q :" tg P AutpXq ; Dg B P AutpBq, τ ˝g " g B ˝τ u, and assume that the image of the homomorphism g P AutpX; τ q Ñ g B P AutpBq is infinite. Then, X is a torus. This result directly follows from the proof of Proposition 3.6 in [START_REF] Cantat | Symétries birationnelles des surfaces feuilletées[END_REF]. In particular the automorphism f B P AutpBq such that π f ˝f " f B ˝πf has finite order when X is a K3, an Enriques, or a rational surface.

EXAMPLES AND CLASSIFICATION

This section may be skipped in a first reading. It describes a few examples, and proves that a compact Kähler surface X is projective when its automorphism group is non-elementary.

3.1. Wehler surfaces (see [START_REF] Cantat | Birational automorphism groups and the movable cone theorem for Calabi-Yau manifolds of Wehler type via universal Coxeter groups[END_REF][START_REF] Reschke | Lower semi-continuity of entropy in a family of K3 surface automorphisms[END_REF][START_REF] Wang | Rational points and canonical heights on K3-surfaces in P 1 ˆP1 ˆP1[END_REF][START_REF] Wehler | K3-surfaces with Picard number 2[END_REF]). Consider the variety M " P 1 ˆP1 ˆP1 and let π 1 , π 2 , and π 3 be the projections on the first, second, and third factor: π i pz 1 , z 2 , z 3 q " z i . Denote by L i the line bundle π i pOp1qq and set

(3.1) L " L 2 1 b L 2 2 b L 2 3 " π 1 pOp2qq b π 2 pOp2qq b π 3 pOp2qq. Since K P 1 " Op´2q,
this line bundle L is the dual of the canonical bundle K M . By definition, |L| » PpH 0 pM, Lqq is the linear system of surfaces X Ă M given by the zeroes of global sections P P H 0 pM, Lq. Using affine coordinates px 1 , x 2 , x 3 q on M " P 1 ˆP1 ˆP1 , such a surface is defined by a polynomial equation P px 1 , x 2 , x 3 q " 0 whose degree with respect to each variable is ď 2 (see [START_REF] Cantat | Dynamique des automorphismes des surfaces K3[END_REF][START_REF] Curtis | Dynamics on K3 surfaces: Salem numbers and Siegel disks[END_REF] for explicit examples). These surfaces will be referred to as Wehler surfaces or (2,2,2)-surfaces; modulo AutpM q, they form a family of dimension 17.

Fix k P t1, 2, 3u and denote by i ă j the other indices. If we project X to P 1 ˆP1 by π ij " pπ i , π j q, we get a 2 to 1 cover (the generic fiber is made of two points, but some fibers may be rational curves). As soon as X is smooth the involution σ k that permutes the two points in each (general) fiber of π ij is an involutive automorphism of X; indeed X is a K3 surface and any birational self-map of such a surface is an automorphism. Proposition 3.1. There is a countable union of proper Zariski closed subsets pW i q iě0 in |L| such that (1) if X is an element of |L|zW 0 , then X is a smooth K3 surface and X does not contain any fiber of the projections π ij ; (2) if X is an element of |L|zp Ť i W i q, the restriction morphism PicpM q Ñ PicpXq is surjective. In particular its Picard number is ρpXq " 3.

See [START_REF] Cantat | Birational automorphism groups and the movable cone theorem for Calabi-Yau manifolds of Wehler type via universal Coxeter groups[END_REF] for the proof of this proposition, as well as that of Lemma 3.2. From the second assertion, we deduce that for a very general X, PicpXq is isomorphic to PicpM q: it is the free Abelian group of rank 3, generated by the classes

(3.2) c i :" rpL i q |X s.
The elements of |pL i q |X | are the curves of X given by the equations z i " α for some α P P 1 .

The arithmetic genus of these curves is equal to 1: in other words the projection pπ i q |X : X Ñ P 1 is a genus 1 fibration. Moreover, for a general choice of X in |L|, pπ i q |X has 24 singular fibers of type I 1 , i.e. isomorphic to a rational curve with exactly one simple double point. The intersection form is given by c 2 i " 0 and xc i |c j y " 2 if i ‰ j, so that its matrix is given by

(3.3) ¨0 2 2 2 0 2 2 2 0 '.
Lemma 3.2. Assume that X does not contain any fiber of the projection π ij . Then, the involution σ k preserves the subspace Zc 1 ' Zc 2 ' Zc 3 of NSpX; Zq and

σ k c i " c i , σ k c j " c j , σ k c k " ´ck `2c i `2c j .
Equivalently, the action of σ k on Vect R pc 1 , c 2 , c 3 q preserves the classes c i and c j and acts as a reflection with respect to the hyperplane Vectpc i , c j q Ă NSpX; Rq. In other words, setting

u k " pc 1 `c2 `c3 q ´2c k , σ k pvq " v `1 2 xv|u k yu k for all v in Zc 1 ' Zc 2 ' Zc 3 .
Combining this lemma with the previous proposition, we see that a very general Wehler surface has Picard number 3, H X has dimension 2, NSpX; Zq " Vect Z pc 1 , c 2 , c 3 q and the matrices of the σ i in the basis pc i q are (3.4)

σ 1 " ¨´1 0 0 2 1 0 2 0 1 ', σ 2 " ¨1 2 0 0 ´1 0 0 2 1 ', σ 3 " ¨1 0 2 0 1 2 0 0 ´1 '. Proposition 3.3.
If X is a very general Wehler surface then:

(1) X is a smooth K3 surface with Picard number 3;

(2) AutpXq is equal to xσ 1 , σ 2 , σ 3 y, it is a free product of three copies of Z{2Z, and AutpXq is a finite index subgroup in the group of integral isometries of NSpX; Zq; (3) AutpXq ˚acts strongly irreducibly on NSpX; Rq;

(4) AutpXq does not preserve any algebraic curve D Ă X;

(5) the limit set of AutpXq ˚is equal to BH X ; (6) the compositions σ i ˝σj and σ i ˝σj ˝σk are respectively parabolic and loxodromic for every triple pi, j, kq with ti, j, ku " t1, 2, 3u.

Proof. The first three assertions follow from Proposition 3.1, [24, §1.5] and [START_REF] Cantat | Birational automorphism groups and the movable cone theorem for Calabi-Yau manifolds of Wehler type via universal Coxeter groups[END_REF]Thm 3.6]. For the fourth one, note that any invariant curve D would yield a non-trivial fixed point rDs in NSpX; Zq, contradicting assertion (3). The fifth one follows from the second because the limit set of a lattice in IsompNSpX; Rqq is always equal to BH X . To prove the last assertion, it suffices to compute the corresponding product of matrices given in Equation (3.4) (see [START_REF] Cantat | Dynamique des automorphismes des surfaces K3[END_REF]).

Remark 3.4. In [START_REF] Baragar | Automorphisms of surfaces in a class of Wehler K3 surfaces with Picard number 4[END_REF], Baragar gives examples of smooth surfaces X P |L| for which ρpXq ě 4 and the limit set of AutpXq ˚in BH X is a genuine fractal set.

3.2. Pentagons. The dynamics on the space of pentagons with given side lengths, introduced in §1.2, shares important similarities with the dynamics on Wehler surfaces. A pentagon with side lengths 0 , . . . , 4 modulo translations of the plane is the same as the data of a 5-tuple of vectors pv i q i"0,...,4 in R 2 (identified with C) of respective length i such that

ř i v i " 0. Write v i " i t i with |t i | " 1.
Then the action of SO 2 pRq can be identified to the diagonal multiplicative action of U 1 " tα P C ; |α| " 1u on the t i :

(3.5) α ¨pt 0 , . . . , t 4 q " pαt 0 , . . . αt 4 q. Now, following Darboux [START_REF] Darboux | De l'emploi des fonctions elliptiques dans la théorie du quadrilatère plan[END_REF], we consider the surface X in P 4 C defined by the equations

(3.6) # 0 z 0 ` 1 z 1 ` 2 z 2 ` 3 z 3 ` 4 z 4 " 0 0 {z 0 ` 1 {z 1 ` 2 {z 2 `
3 {z 3 ` 4 {z 4 " 0 where rz 0 : . . . : z 4 s is some fixed choice of homogeneous coordinates, and the second equation must be multiplied by z 0 z 1 z 2 z 3 z 4 to obtain a homogeneous equation of degree 4.

Remark 3.5. This surface is isomorphic to the Hessian of a cubic surface (see [53, §9]). More precisely, consider a cubic surface S Ă P 3 C whose equation F can be written in Sylvester's pentahedral form that is, as a sum F " ř 4 i"0 λ i F 3 i for some complex numbers λ i and linear forms F i with ř 4 i"0 F i " 0. By definition, its Hessian surface H F is defined by detpB i B j F q " 0. Then, using the linear forms F i to embed H F in P 4 C , we obtain the surface defined by the pair of equations ř 4 i"0 z i " 0 and

ř 4 i"0 1 λ i z i " 0.
Thus, H F is our surface X, for 2 i " λ i . We refer to [START_REF] Dolgachev | Birational automorphisms of quartic Hessian surfaces[END_REF][START_REF] Dardanelli | Hessians and the moduli space of cubic surfaces[END_REF][START_REF] Dolgachev | Salem numbers and Enriques surfaces[END_REF][START_REF] Rosenberg | Hessian quartic surfaces that are Kummer surfaces[END_REF] for an introduction to these surfaces and their birational transformations. Lemma 3.6. Let " p 0 , . . . , 4 q be an element of pC ˚q5 . The surface X Ă P 4

C defined by the system (3.6) has 10 singularities at the points q ij determined by the system of equations i z i ` j z j " 0, z k " z l " z m " 0 with i ă j and ti, j, k, l, mu " t0, 1, 2, 3, 4u. In the complement of these ten isolated singularities, X is smooth if and only if

(3.7) 4 ÿ i"0 ε i i ‰ 0 @ε i P t˘1u .
The proof is elementary and left to the reader. Lemma 3.7. If P pC ˚q5 satisfies Condition (3.7), then the ten singularities are simple nodes (Morse singularities) and the surface X is a (singular) K3 surface: a minimal resolution X of X is a K3 surface, which is obtained by blowing-up its ten nodes, thereby creating ten rational p´2q-curves.

Proof. Working in the chart z 0 " 1 and replacing z 4 by ´p 0 ` 1 z 1 ` 2 z 2 ` 3 z 3 q{ 4 , the quadratic term of the equation of X at the singularity pz 1 , z 2 , z 3 q " p0, 0, 0q is p´ 0 { 4 qQ, where

(3.8) Qpz 1 , z 2 , z 3 q " 1 z 2 z 3 ` 2 z 1 z 3 ` 3 z 1 z 2
is a non-degenerate quadratic form (its determinant is 2 1 2 3 ‰ 0). So locally X is holomorphically equivalent to the quadratic cone tQ " 0u, hence to a quotient singularity pC 2 , 0q{η with ηpx, yq " p´x, ´yq. The minimal resolution of such a singularity is obtained by a simple blow-up of the ambient space, the exceptional divisor being a p´2q-curve in the smooth surface X. The adjunction formula shows that there is a holomorphic 2-form Ω X on the regular part of X; locally, Ω X lifts to an η-invariant form Ω 1 X on C 2 zt0u, which by Hartogs extends at the origin to a non-vanishing 2-form. To recover X, one can first blow-up C 2 at the origin and then take the quotient by (the lift of) η: a simple calculation shows that Ω 1 X determines a non-vanishing 2-form on X. After such a surgery is done at the ten nodes, X is a smooth surface with a non-vanishing section of K X ; since it contains at least ten rational curves, it can not be an Abelian surface, so it must be a K3 surface. Remark 3.8. Let L ij be the line defined by the equations z i " 0, z j " 0, 0 z 0 `¨¨¨` 4 z 4 " 0; each of these ten lines is contained in X, each of them contains 3 singularities of X (namely q kl , q lm , q km with obvious notations), and each singularity is contained in three of these lines. If one projects them on a plane, the ten lines L ij form a Desargues configuration (see [START_REF] Dolgachev | Salem numbers and Enriques surfaces[END_REF][START_REF] Dolgachev | Birational automorphisms of quartic Hessian surfaces[END_REF]).

All this works for any choice of complex numbers i ‰ 0. Now, since the i are real, X is endowed with two real structures. First, one can consider the complex conjugation c : rz i s Þ Ñ rz i s on P 4 pCq and restrict it to X: this gives a first antiholomorphic involution c X . Another one is given by s X : rz i s Þ Ñ r1{z i s. To be more precise, consider first, the quartic birational involution J P BirpP 4 C q defined by Jprz i sq " r1{z i s; J preserves X, it determines a birational transformation J X P BirpXq, and on X it becomes an automorphism because every birational transformation of a K3 surface is regular. Thus, s X " J X ˝cX determines a second antiholomorphic involution s X of X. In what follows, we denote by pX, s X q this real structure (even if it would be better to study it on X); its real part is the fixed point set of s X , i.e. the set of points in XpCq with coordinates of modulus 1: the real part does not contain any of the singularities of X, this is why we prefer to stay in X rather than lift everything to X. Thus, with the real structure defined by s X , the real part of X coincides with Pent 0 p 0 , . . . , 4 q if p i q P pR ˚q5 . Remark 3.9. When i ą 0 for all indices i P t0, . . . , 4u, a complete description of the possible homeomorphism types for the real locus (in the smooth and singular cases) is given in [START_REF] Curtis | Configuration spaces of planar pentagons[END_REF]: in the smooth case, it is an orientable surface of genus g " 0, . . . , 4 or the union of two tori.

Remark 3.10. The involution J preserves X and the two real structures pX, c X q and pX, s X q. It lifts to a fixed point free involution ĴX on X, and X{ ĴX is an Enriques surface. On pentagons, J corresponds to the symmetry px, yq P R 2 Þ Ñ px, ´yq that reverses orientation. Thus we see that the space of pentagons modulo affine isometries is an Enriques surface. When X acquires an eleventh singularity which is fixed by J X , then X{ ĴX becomes a Coble surface: see [51, §5] for nice explicit examples. This happens for instance when all lengths are 1, except one which is equal to 2 (this corresponds to t " 1{4 in [51, §5.2]).

Finally, let us express the folding transformations in coordinates. Given i ‰ j in t0, . . . , 4u (consecutive or not) we define an involution pt i , t j q Þ Ñ pt 1 i , t 1 j q preserving the vector i t i ` j t j by taking the symmetric of t i and t j with respect to the line directed by i t i ` j t j . In coordinates, t 1 k " u{t k for some u of modulus 1, and equating i t i ` j t j " i t 1 i ` j t 1 j one obtains (3.9)

pt 1 i , t 1 j q " ˆu t i , u t j ˙, with u " i t i ` j t j i t ´1 i ` j t ´1 j .
Observe that these computations also make sense when the i are complex numbers, or when we replace the t i by the complex numbers z i . This defines a birational involution σ ij : X X,

(3.10) σ ij rz 0 : . . . : z 4 s " rz 1 0 : . . . :

z 1 4 s with z 1 k " z k if k ‰ i, j, z 1
i " vz j , and z 1 j " vz i with v " p i z i ` j z j q{p i z j ` j z i q. Again, since every birational self-map of a K3 surface is an automorphism, these involutions σ ij are elements of Autp Xq that commute with the antiholomorphic involution s X ; hence, they generate a subgroup of Autp X; s X q. Thus we have constructed a family of projective surfaces X, depending on a parameter P P 4 pCq, endowed with a group of automorphisms generated by involutions. Note that this group can be elementary: for instance when the five lengths are all equal the group is finite because in that case pz 1 i , z 1 j q " pz j , z i q. When j " i `1 modulo 5, σ ij corresponds to the folding transformation described in the introduction.

Remark 3.11. Pick a singular point q ij , and project X from that point onto a plane, say the plane tz i " 0u in the hyperplane P " t 0 z 0 `¨¨¨` 4 z 4 " 0u. One gets a 2 to 1 cover X Ñ P 2 C , ramified along a sextic curve (this curve is the union of two cubics, see [START_REF] Rosenberg | Hessian quartic surfaces that are Kummer surfaces[END_REF]). The involution σ ij permutes the points in the fibers of this 2 to 1 cover: if x is a point of X, the line joining q ij and x intersects X in the third point σ ij pxq. The singularity q ij is an indeterminacy point, mapped by σ ij to the opposite line L ij . Proposition 3.12. For a general parameter P P 4 pCq:

(1) X is a K3 surface with ten nodes, with two real structures c X and s X when P P 4 pRq;

(2) if i, j " i `1, k " i `2 are distinct consecutive indices (modulo 5), then σ ij ˝σjk is a parabolic transformation on X; (3) if i, j, k, and l are four distinct indices (modulo 5), then σ ij commutes to σ kl . (4) the group Γ generated by the involutions σ ij is a non-elementary subgroup of Autp X; s X q that does not preserve any algebraic curve.

In [START_REF] Dolgachev | Salem numbers and Enriques surfaces[END_REF], Dolgachev computes the action of σ ij on NSp Xq. This contains a proof of this proposition. He also describes, up to finite index, the Coxeter group generated by the σ ij . The automorphism groups of X and of the Enriques surface X{ ĴX are described in [START_REF] Dolgachev | Birational automorphisms of quartic Hessian surfaces[END_REF] and [START_REF] Shimada | On an Enriques surface associated with a quartic Hessian surface[END_REF]. [START_REF] Franc | Enriques surfaces. I[END_REF][START_REF] Dolgachev | A brief introduction to Enriques surfaces[END_REF]). Enriques surfaces are quotients of K3 surfaces by fixed point free involutions. According to Horikawa and Kondō ([75,[START_REF] Horikawa | On the periods of Enriques surfaces[END_REF][START_REF] Kondō | The rationality of the moduli space of Enriques surfaces[END_REF]), the moduli space M E of complex Enriques surfaces is a rational quasi-projective variety of dimension 10. An Enriques surface X is nodal if it contains a smooth rational curve; such rational curves have self-intersection ´2, and are called nodal-curves or p´2q-curves. Nodal Enriques surfaces form a hypersurface in M E .

Enriques surfaces (see

For any Enriques surface X, the lattice pNSpX; Zq, q X q is isomorphic to the orthogonal direct sum E 10 " U k E 8 p´1q, ( 1 ). Let W X Ă OpNSpX; Zqq be the subgroup generated by reflexions about classes u such that u 2 " ´2, and W X p2q be the subgroup of W X acting trivially on NSpX; Zq modulo 2. Both W X and W X p2q have finite index in OpNSpX; Zqq. The following result is due independently to Nikulin and Barth and Peters (see [START_REF] Dolgachev | A brief introduction to Enriques surfaces[END_REF] for details and references).

Theorem 3.13. If X is an Enriques surface which is not nodal, the homomorphism AutpXq Q f Þ Ñ f ˚P GLpH 2 pX, Zqq is injective, and its image satisfies W X p2q Ă AutpXq ˚Ă W X .
In particular, for any unnodal Enriques surface, AutpXq is non-elementary, contains parabolic elements, and acts irreducibly on NSpX; Rq; thus, it does not preserve any curve.

3.4.

Examples on rational surfaces: Coble and Blanc. Closely related to Enriques surfaces are the examples of Coble, obtained by blowing up the ten nodes of a general rational sextic curve C 0 Ă P 2 . The result is a rational surface X with a large group of automorphisms. To be precise, consider the canonical class K X Ă NSpX; Zq; its orthogonal complement K K X is a lattice of dimension 10, isomorphic to E 10 , and we define W X p2q exactly in the same way as for Enriques surfaces. Then, AutpXq ˚preserves the decomposition K X ' K K X , and AutpXq contains W X p2q when X does not contain any smooth rational curve of self-intersection ´2 (see [START_REF] Cantat | Rational surfaces with a large group of automorphisms[END_REF], Theorem 3.5). Also, Coble surfaces may be thought of as degeneracies of Enriques surfaces: an interesting difference is that rK X s is non trivial; in particular, NSpX; Zq 0 is always non-trivial, for any Γ Ă AutpXq. There is a holomorphic section of ´2K X vanishing exactly along the strict transform C Ă X of the rational sextic curve C 0 ; this means that there is a meromorphic section Ω X " ξpx, yqpdx ^dyq 2 of K b2 X that does not vanish and has a simple pole along C. Thus, the formula

(3.11) vol X pU q " ż U |ξpx, yq| dx ^dy ^dx ^dy " ż U |ξpx, yq| pidx ^dxq ^pidy ^dyq determines a finite measure( 2 ) vol X " " Ω 1{2 X

^Ω1{2

X ", which we may assume to be a probability after multiplying Ω X by some adequate constant; this measure is AutpXq-invariant (because vol X is uniquely determined by the complex structure).

Another family of examples has been described by Blanc in [START_REF] Blanc | On the inertia group of elliptic curves in the Cremona group of the plane[END_REF]. One starts with a smooth cubic curve C 0 Ă P 2 . If q 1 is a point of C 0 , there is a unique birational involution s 1 of P 2 that fixes C 0 pointwise and preserves the pencil of lines through q 1 . The indeterminacy points of s 1 are q 1 and the four tangency points of C 0 with this pencil (one of them may be "infinitely near q 1 " and in that case it corresponds to the tangent direction of C 0 at q 1 ); thus the indeterminacies of s 1 are resolved by blowing-up points of C 0 (or points of its strict transform). After such a sequence of blow-ups s 1 becomes an automorphism of a rational surface X 1 that fixes pointwise the strict transform of C 0 . So, if we blow-up other points of this curve, s 1 lifts to an automorphism of the 1 Here, U is the standard 2-dimensional Minkowski lattice, pZ 2 , x1x2q, and E8 is the root lattice given by the corresponding Dynkin diagram; so E8p´1q is negative definite, and E10 has signature p1, 9q (see [START_REF] Franc | Enriques surfaces. I[END_REF]Chap. II]). Also, recall that in this paper NSpX; Zq denotes the torsion free part of the Néron-Severi group, which is sometimes denoted by NumpX; Zq in the literature on Enriques surfaces.

2 if locally C " tx " 0u then ξpx, yq " ηpx, yq{x where η is regular; thus, |ξ| " |η| |x| ´1 is locally integrable because 1 r α is integrable with respect to rdrdθ when α ă 2 new surface. In particular, we can start with a finite number of points q i P C 0 , i " 1, . . . , k, and resolve simultaneously the indeterminacies of the involutions s i determined by the q i . The result is a surface X, with a subgroup Γ :" xs 1 , . . . , s k y of AutpXq. Blanc proves that (1) there are no relations between these involutions, that is, Γ is a free product xs 1 , . . . , s k y » ˚k i"1 Z{2Z, (2) the composition of two distinct involutions s i ˝sj is parabolic, and (3) the composition of three distinct involutions is loxodromic. There is a meromorphic section Ω X of K X with a simple pole along the strict transform of C 0 , but the form vol X :" Ω X ^ΩX is not integrable.

3.5. Real forms. For each of the examples described in Sections 3.1 to 3.4, we may ask for the existence of an additional real structure on X, and look at the group of automorphisms AutpX R q that preserve the real structure (automorphisms commuting with the anti-holomorphic involution describing the real structure). Note that if X is a smooth projective variety with a real structure, then XpRq is either empty or a compact, smooth, and totally real surface in X.

If X is a Wehler surface defined by a polynomial equation P px 1 , x 2 , x 3 q with real coefficients the σ i are automatically defined over R. If X is a Blanc surface for which C 0 is defined over R and the points q i are chosen in C 0 pRq, then again xs 1 , . . . , s k y Ă AutpX R q. Real Enriques and Coble surfaces provide also many examples for which AutpX R q is non-elementary (see [START_REF] Degtyarev | Real Enriques surfaces[END_REF]).

3.6. Surfaces admitting non-elementary groups of automorphisms. All surfaces in the previous examples are projective. This is a general fact, which we prove in this paragraph: we rely on the Kodaira-Enriques classification to describe compact Kähler surfaces which support a non-elementary group of automorphisms and prove Theorem E.

3.6.1. Minimal models. We refer to Theorem 10.1 of [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF] for the following result: Theorem 3.14. If X is a compact Kähler surface with a loxodromic automorphism, then either X is a rational surface, and there is a birational morphism π : X Ñ P 2 C ; -or the Kodaira dimension of X is equal to 0, and there is an AutpXq-equivariant bimeromorphic morphism π : X Ñ X 0 such that X 0 is a compact torus, a K3 surface, or an Enriques surface.

In particular, h 2,0 pXq equals 0 or 1.

Remark 3.15. If X is a torus or K3 surface, there is a holomorphic 2-form Ω X on X that does not vanish and satisfies ş X Ω X ^ΩX " 1. It is unique up to multiplication by a complex number of modulus 1. A consequence of utmost importance to us is that the volume form

(3.12) Ω X ^ΩX
is AutpXq-invariant. Furthermore for every f we can write f ˚ΩX " Jpf qΩ X , where the Jacobian f P AutpXq Þ Ñ Jpf q P U 1 is a unitary character on the group AutpXq. Since H 2,0 pX; Cq is generated by rΩ X s, we obtain f ˚w " Jpf qw @w P H 2,0 pX; Cq. If Y is an Enriques surface, and X Ñ Y is its universal cover, then X is a K3 surface: the volume form Ω X ^ΩX is invariant under the group of deck transformations, and determines an AutpY q-invariant volume form on Y . So, if X is not rational, the dynamics of AutpXq is conservative: it preserves a canonical volume form which is uniquely determined by the complex structure of X.

It follows from Theorem 3.14 that, in most cases, AutpXq is countable (see [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF]Rmk 3.3]).

Proposition 3.16. Let X be a compact Kähler surface. If AutpXq contains a loxodromic element, then the kernel of the homomorphism AutpXq Ñ AutpXq ˚Ă GLpNSpX; Zqq is finite unless X is a torus. So, if AutpXq is non-elementary, then AutpXq is discrete or X is a torus.

3.6.2. Projectivity.

Theorem E. Let X be a compact Kähler surface and Γ be a non-elementary subgroup of AutpXq. Then X is projective, and is birationally equivalent to a rational surface, an Abelian surface, a K3 surface, or an Enriques surface.

From the discussion in § §3.1-3.4 we see that there exist examples with a non-elementary group of automorphisms for each of these four classes of surfaces. Theorem E is a direct consequence of Theorem 3.14 and the following lemmas. Lemma 3.17. Let f be a loxodromic automorphism of a compact Kähler surface X. The following properties are equivalent:

(1) on H 2,0 pX; Cq, f ˚acts by multiplication by a root of unity;

(2) X is projective.

Proof of Lemma 3.17. The characteristic polynomial χ f of f ˚: H 2 pX; Zq Ñ H 2 pX; Zq is a monic polynomial with integer coefficients. Since f is loxodromic, f ˚has a real eigenvalue λpf q ą 1. Besides λpf q and λpf q ´1, all other roots of χ f have modulus 1, so λpf q is a reciprocal quadratic integer or a Salem number (see § 2.4.3 of [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF] for more details). Thus, the decomposition of χ f into irreducible factors can be written as

(3.13) χ f ptq " S f ptq ˆRf ptq " S f ptq ˆm ź i"1 C f,i ptq
where S f is a Salem polynomial or a reciprocal quadratic polynomial, and the C f,i are cyclotomic polynomials. In particular if ξ is an eigenvalue of f ˚and a root of unity, we see that ξ is a root of R f ptq but not of S f ptq.

The subspace H 2,0 pCq Ă H 2 pX; Cq is f ˚-invariant and, by Lemma 2.1, all eigenvalues of f ˚on that subspace have modulus 1; if an eigenvalue of f ˚|H 2,0 pX;Cq is not a root of unity, then it is a root of S f .

Assume that all eigenvalues of f ˚on H 2,0 pX; Cq are roots of unity. Then KerpS f pf ˚qq Ă H 2 pX; Rq is a f ˚-invariant subspace of H 1,1 pX; Rq. This subspace is defined over Q and is of Minkowski type; in particular, it contains integral classes of positive self-intersection, and by the Kodaira embedding theorem, X is projective. Conversely, assume that X is projective. The Néron-Severi group NSpX; Qq Ă H 1,1 pX; Rq is f ˚-invariant and contains vectors of positive self-intersection, so by Proposition 2.8 it contains all isotropic lines associated to loxodromic automorphisms. Now any f ˚invariant subspace defined over Q and containing the eigenspace associated to λpf ˚q contains KerpS f pf ˚qq, so we deduce that KerpS f pf ˚qq Ă NSpX; Qq. In particular, KerpS f pf ˚qq does not intersect H 2,0 pX; Cq, which is invariant, and we conclude that all eigenvalues of f ˚on H 2,0 pX; Cq are roots of unity. Lemma 3.18. Let X be a compact Kähler surface. If X is not projective, then AutpXq ˚is virtually Abelian and if it contains a loxodromic element it is virtually cyclic.

Proof. Assume that AutpXq ˚is not virtually Abelian, or that it contains a loxodromic element without being virtually cyclic. According to Theorem 3.2 of [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF], AutpXq ˚contains a non-Abelian free group Γ such that all elements of Γztidu are loxodromic; from Theorem 3.14, either h 2,0 pXq " 0 or X is the blow-up of a torus or a K3 surface. In the first case, H 2 pX; Rq " H 1,1 pX; Rq so, by the Hodge index theorem, H 1,1 pX; Rq contains an integral class with positive self-intersection; then, the Kodaira embedding theorem shows that X is projective. In the second case, by uniqueness of the minimal model, the morphism X Ñ X 0 onto the minimal model of X is AutpXq-equivariant, so we can assume that X " X 0 is minimal and h 2,0 pXq " 1. Consider the homomorphism J : AutpXq Ñ U 1 , as in Remark 3.15. Since U 1 is Abelian kerpJ| Γ q contains loxodromic elements: indeed if f, g P Γ and f ‰ g then rf, gs " f gf ´1g ´1 is loxodromic and Jprf, gsq " 1. From Lemma 3.17 we deduce that X is projective.

GLOSSARY OF RANDOM DYNAMICS, I

We now initiate the random iteration by introducing a probability measure on AutpXq. In this section we introduce a first set of ideas from the theory of random dynamical systems, as well as some notation that will be used throughout the paper. 4.1. Random holomorphic dynamical systems. Let X be a compact Kähler surface, such that AutpXq is non-elementary. Note that AutpXq is locally compact for the topology of uniform convergence -in many interesting cases it is actually discrete (see Proposition 3.16)-so it admits a natural Borel structure. We fix some Riemannian structure on X, for instance the one induced by the Kähler form κ 0 . For f P AutpXq, we denote by }f } C 1 the maximum of }Df x } where the norm of Df x : T x M Ñ T f pxq M is computed with respect to this Riemannian metric.

We consider a probability measure ν on AutpXq satisfying the moment condition (or integrability condition) (4.1)

ż ´log }f } C 1 pXq `log › › f ´1› › C 1 pXq ¯dνpf q ă `8.
The finiteness of the integral in (4.1) does not depend on our choice of Riemannian metric. When the support of ν is finite, the integrability (4.1), as well as stronger moment conditions which will appear later (see Conditions (5.23) and (5.24)), are obviously satisfied.

Lemma 4.1. The measure ν satisfies the moment condition (4.1) if and only if it satisfies the higher moment conditions

ż ´log }f } C k pXq `log › › f ´1› › C k pXq ¯dνpf q ă 8, for all k ě 1.
This lemma follows from the Cauchy estimates. In particular, if ν satisfies (4.1), then it satisfies a similar moment condition for the C 2 norm, a property required to apply Pesin's theory.

Given ν, we shall consider independent, identically distributed sequences pf n q ně0 of random automorphisms of X with distribution ν, and study the dynamics of random compositions of the form f n´1 ˝¨¨¨˝f 0 . The data pX, νq will be referred to as a random holomorphic dynamical system on X. Many properties of pX, νq depend on the properties of the subgroup (4.2)

Γ " Γ ν :" xSupppνqy generated by (the support of) ν in AutpXq. If in addition Γ ν is non-elementary, we say that pX, νq is non-elementary.

4.2.

Invariant and stationary measures. Let G be a topological group and ν be a probability measure on G. Consider a measurable action of G on some measurable space pM, Aq. Every f P G determines a push-forward operator µ Þ Ñ f › µ, acting on positive (resp. probability) measures µ on pM, Aq. By definition, a probability measure µ on pM, Aq is ν-stationary if

(4.3) ż f › µ dνpf q " µ,
and it is ν-almost surely invariant if f › µ " µ for ν-almost every f . Let us stress that we only deal with probability measures in this definition; slightly abusing terminology, most often we drop the mention to ν and the mention that µ is a probability. A stationary measure is ergodic if it is an extremal point of the closed convex set of stationary measures (see [12, §2.1.3]).

If µ is almost surely invariant then it is stationary but the converse is generally false. If M is compact, the action G ˆM Ñ M is continuous, and A is the Borel σ-algebra, the Kakutani fixed point theorem implies the existence of at least one stationary measure. On the other hand the existence of an invariant measure is a very restrictive property (see Sections 1.3 and 5.3). Following Furstenberg [START_REF] Furstenberg | Stiffness of group actions[END_REF] we say that an action is stiff (or ν-stiff) if any ν-stationary measure is ν-almost surely invariant.

We shall consider several measurable actions of AutpXq: its tautological action on X, but also its action on the projectivized tangent bundle PpT Xq, on cohomology groups of X and their projectivizations, on spaces of currents, etc. In all cases, M will be a locally compact space and A its Borel σ-algebra, which will be denoted by BpM q. Remark 4.2. Since X is compact and the action AutpXq ˆX Ñ X is continuous, a probability measure µ on pX, BpXqq is ν-almost surely invariant if and only if it is invariant under the action of the closure of Γ ν in AutpXq; this follows from the dominated convergence theorem.

Random compositions.

Set Ω " AutpXq N , endowed with its product topology. The associated Borel σ-algebra coincides with the product σ-algebra, and it is generated by cylinders (see § 7.1). We endow Ω with the product measure ν N . Choosing a random element in Ω with respect to ν N is equivalent to choosing an independent and identically distributed random sequence of automorphisms in AutpXq with distribution ν. For ω P Ω, we let f ω " f 0 and denote by f n ω the left composition of the n first terms of ω, that is (4.4)

f n ω " f n´1 ˝¨¨¨˝f 0 for n ą 0. By definition f 0 ω " id. Let us record for future reference the following consequence of the Borel-Cantelli lemma. We denote by σ : Ω Ñ Ω the unilateral shift. Lemma 4.3. If pX, νq is a random dynamical system satisfying the moment condition (4.1), then for ν N -almost every sequence ω " pf n q P Ω,

1 n `log }f n } C 1 `log › › f ´1 n › › C 1 ˘ÝÑ nÑ8 0.

FURSTENBERG THEORY IN H 1,1 pX; Rq

Consider a non-elementary random holomorphic dynamical system pX, νq on a compact Kähler surface, satisfying the moment condition (4.1). The main purpose of this section is to analyze the linear action of pX, νq on H 1,1 pX, Rq by way of the theory of random products of matrices. Basic references for this subject are the books by Bougerol and Lacroix [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF] and by Benoist and Quint [START_REF] Benoist | Random walks on reductive groups, volume 62 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]. 5.1. Moments and cohomology. Let M be a compact connected manifold of dimension m, endowed with some Riemannian metric g. If f : M Ñ M is a smooth map, }f } C 1 denotes the maximum norm of its tangent action, computed with respect to g (see Section 4.1). Thus, f is a Lipschitz map with Lippf q " }f } C 1 for the distance determined by g; in particular }f } C 1 ě 1 whenever f is onto. Fix a norm |¨| H k on each cohomology group H k pM ; Rq, for 0 ď k ď m. Lemma 5.1. There is a constant C ą 0, that depends only on M , g, and the norms

|¨| H k , such that |f ˚rαs| H k ď C k Lippf q k |rαs| H k for every class rαs P H k pM ; Rq and every map f : M Ñ M of class C 1 .
In other words, the operator norm }f ˚}H k is controlled by the Lipschitz constant:

}f ˚}H k ď C k Lippf q k ď C k }f } k C 1 .
Proof. Pick a basis of the homology group H k pM ; Rq » H k pM ; Rq ˚given by smoothly immersed, compact, k-dimensional manifolds ι i : N i Ñ M , and a basis of H k pM ; Rq given by smooth k-forms α j . Then, the integral

ş N i ι i pf ˚αj q is bounded from above by C k }f } k C 1 for some constant C, because (5.1) |pf ˚αj q x pv 1 , . . . , v k q| " |α j pf ˚v1 , . . . , f ˚vk q| ď c j }f } k C 1 k ź "1 |v | g
for every point x P M and every k-tuple of tangent vectors v P T x M ; here, c j is the supremum of the norm of the multilinear map pα j q x over x P M .

If ν is a probability measure on DiffpM q satisfying the moment condition (4.1), then

(5.2) @1 ď k ď m, ż DiffpM q log p}f ˚}H k q `log `› › pf ´1q ˚› › H k ˘dνpf q ă `8.
If we specialize this to automorphisms of compact Kähler surfaces we get

(5.3) ż AutpXq log p}f ˚}H 1,1 q `log `› › pf ´1q ˚› › H 1,1 ˘dνpf q ă `8,
which is actually equivalent to (5.2) by Lemma 2.2. We saw in §2.3.

3 that }f ˚}H 1,1 - › › pf ´1q ˚› › H 1,1 , so this last condition is in turn equivalent to (5.4) ż AutpXq log p}f ˚}H 1,1 q dνpf q ă `8.

Cohomological Lyapunov exponent.

From now on we denote by |¨| a norm on H 1,1 pX, Rq and by }¨} the associated operator norm. The linear action induced by the random dynamical system pX, νq on H 1,1 pX, Rq defines a random product of matrices. Since the moment condition (5.4) is satisfied, we can define the upper Lyapunov exponent λ H 1,1 (or λ H 1,1 pνq) by

λ H 1,1 " lim nÑ`8 1 n ż logp}pf n ω q ˚}qdν N pωq (5.5) " lim nÑ`8 1 n log }pf n ω q ˚} (5.6)
where the second equality holds almost surely, i.e. for ν N -almost every ω P Ω. This convergence follows from Kingman's subadditive ergodic theorem, since }¨} being an operator norm, pω, nq Þ Ñ logp}pf n ω q ˚}q defines a subadditive cocycle (see [START_REF] Benoist | Random walks on reductive groups, volume 62 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]Thm 4.28] or [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF]Thm I.4.1]). Note that pf n ω q ˚" f 0 ˝¨¨¨˝f n´1 , so we are dealing with right compositions instead of the usual left composition. However since f 0 ˝¨¨¨˝f n´1 has the same distribution as f n´1 ˝¨¨¨˝f 0 , the Lyapunov exponent in (5.5) corresponds to the usual definition of the upper Lyapunov exponent of the random product of matrices. We refer to [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF][START_REF] Ledrappier | Quelques propriétés des exposants caractéristiques[END_REF] for the definition and main properties of the subsequent Lyapunov exponents (see also [12, §10.5]). Proposition 5.2. Let pX, νq be a non-elementary holomorphic dynamical system on a compact Kähler surface, satisfying the moment condition (4.1), or more generally (5.4). Then the cohomological Lyapunov exponent λ H 1,1 is positive and the other Lyapunov exponents of the linear action on H 1,1 pX, Rq are ´λH 1,1 , with multiplicity 1, and 0, with multiplicity h

1 n log |pf n ω q ˚a| " λ H 1,1
for ν N -almost every ω.

Proof. Corollary 2.5 implies that if a P H X then for every f P AutpXq, |f ˚a| -}f ˚}, where the implied constants depend only on a. Thus the result follows from Equation (5.6).

Remark 5.4. It is natural to expect that Lemma 5.3 holds for any a P Π Γ z t0u; this is true under the more stringent moment assumption (5.23) (see the proof of Proposition 5.14 below).

If the order of compositions is reversed (which is less natural from the point of view of iterated pull-backs), then Lemma 5.3 indeed holds for any a in Π Γν (see [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF]Cor. III.3.4.i]): Lemma 5.5. For any a P Π Γν z t0u and for ν N -almost every ω " pf n q ně0 P Ω we have

lim nÑ`8 1 n log |f n ¨¨¨f 1 a| " λ H 1,1 .

5.3.

The measure µ B . By Furstenberg's theory the linear projective action of the random dynamical system pX, νq on PΠ Γν Ă PH 1,1 pX; Rq admits a unique stationary measure µ PΠ Γν ; this measure does not charge any proper projective subspace of PΠ Γν . Recall that the mass of a class a is defined by Mpaq " xa|rκ 0 sy (see § 2.2).

Lemma 5.6. For ν N -almost every ω, there exists a unique nef class epωq such that Mpepωqq " 1 and

(5.7) 1 Mppf n ω q ˚aq pf n ω q ˚a ÝÑ nÑ8 epωq for any pseudo-effective class a with a 2 ą 0 (in particular for any Kähler class). In addition, the class epωq is almost surely isotropic and Ppepωqq is a point of the limit set LimpΓ ν q Ă BH X .

Before starting the proof, note that Γ ν | Π Γν is proximal, in the sense of [12, §4.1]; equivalently, Γ ν | Π Γν is contracting, in the sense of [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF]Def III.1.3]. In other words, there are sequences of elements g n P Γ ν such that }g n} ´1g n| Π Γν converges to a matrix of rank 1: for instance one can take g n " f n , where f P Γ ν is any loxodromic automorphism.

Proof. For f P AutpXq, we use the notation f ˚for its action on PH 1,1 pX; Rq. Since the action of Γ ν on Π Γν is strongly irreducible and proximal, its projective action satisfies the following contraction property (see [18, Thm III.3.1]): there is a measurable map ω P Ω Þ Ñ epωq P PΠ Γν such that for almost every ω, any cluster value Lpωq of (5.8)

1 }f 0 ¨¨¨f n } f 0 ¨¨¨f n
in EndpΠ Γν q is an endomorphism of rank 1 whose range is equal to Repωq.

Let epωq be the unique vector of mass 1 in the line Repωq. If a P Π Γν satisfies a 2 ą 0 and Mpaq ą 0, then any cluster value of Mppf n ω q ˚aq ´1pf n ω q ˚a must coincide with epωq because by Corollary 2.5 the mass Mppf n ω q ˚aq is comparable to the norm }f 0 ¨¨¨f n }. Thus, the convergence (5.7) is satisfied. Furthermore epωq is nef, because we can apply this convergence to a nef class a and AutpXq preserves the nef cone. Also, epωq belongs to LimpΓ ν q, hence it is isotropic. Now, let a and a 1 be two classes of H X with a P Π Γν . Since the hyperbolic distance between pf n ω q ˚paq and pf n ω q ˚pa 1 q remains constant and the convergence (5.7) holds for a, it also holds for a 1 . This concludes the proof, for every class with positive self-intersection is proportional to a unique class in H X .

Here is a summary of the properties of the stationary measure µ PΠ Γν ; from now on, we view it as a measure on PH 1,1 pX; Rq and rename it as µ B because it is supported on BH X .

Theorem 5.7. The probability measure defined on PH 1,1 pX; Rq by (5.9) µ B " ż δ Ppepωqq dν N pωq is ν-stationary and ergodic. It is the unique stationary measure on PH 1,1 pX; Rq such that µ B pPpΠ K Γν qq " 0. The measure µ B has no atoms and is supported on LimpΓ ν q; in particular, if

Λ 1 Ă LimpΓ ν q is such that µ B pΛ 1 q ą 0 then Λ 1 is uncountable.
The top Lyapunov exponent satisfies the so-called Furstenberg formula:

λ H 1,1 " ż log ˆ|f ˚ũ| |ũ| ˙dνpf q dµ B puq, (5.10)
where ũ P H 1,1 pX, Rqz t0u denotes any lift of u P LimpΓ ν q Ă PH 1,1 pX, Rq.

Proof. The ergodicity of µ B " µ PΠ Γν as well as its representation (5.9) follow from the properties of the action of Γ ν on PpΠ Γ q (see [18, Chap. III]). Also, we know that λ H 1,1 is equal to the top Lyapunov exponent of the restriction of the action to PpΠ Γν q, so the formula (5.10) follows from the strongly irreducible case (see [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF]Cor III.3.4]).

Let now µ be a stationary measure on PH 1,1 pX; Rq such that µpPΠ K Γν q " 0. A martingale convergence argument shows that pf n ω q ˚µ converges to some measure µ ω for almost every ω (see [18, Lem. II.2.1]). Since Γ ν preserves the decomposition Π Γν ' Π K Γν and }pf n ω q ˚} tends to infinity while pf n ω q ˚|Π K Γν stays uniformly bounded, we get that pf n ω q ˚u converges to PΠ Γν for µ-almost every u and ν N -almost every ω; thus µ ω is almost surely supported on PΠ Γν . Since by stationarity µ " ş µ ω dν N pωq we conclude that µ gives full mass to PpΠ Γν q, hence µ " µ B .

Remark 5.8. If Supppνq generates Γ ν as a semi-group, then Supppµ B q " LimpΓ ν q, otherwise the inclusion can be strict: take a Schottky group Γ " xf, gy Ă PSLp2, Rq and ν " pδ f `δg q{2.

Remark 5.9. Since LimpΓ ν q Ă PsefpXq, for every u P LimpΓ ν q there exists a unique ũ such that Pũ " u and xũ | rκ 0 sy " Mpũq " 1. Then the following formula holds:

λ H 1,1 " ż log pMpf ˚ũqq dνpf q dµ B puq " ż log ˆMpf ˚ũq Mpũq ˙dνpf q dµ B puq. (5.11)
Indeed set rpwq " Mpwq{ |w|. On the limit set this function satisfies 1{C ď rpũq ď C, where C is the positive constant from Equation (2.4). Then, for all m ě 1,the stationarity of µ B implies ż log ˆrpf ˚ũq rpũq ˙dνpf q dµ B puq " ż log ˆrpf m ¨¨¨f 0 ũq rpf m´1 ¨¨¨f 0 ũq ˙dνpf m q ¨¨¨dνpf 0 q dµ B puq.

Summing from m " 0 to n ´1, telescoping the sum, and dividing by n gives

ż log ˆrpf ˚ũq rpũq ˙dνpf q dµ B puq " 1 n ż log ˆrpf n´1 ¨¨¨f 0 ũq rpũq ˙dνpf n´1 q ¨¨¨dνpf 0 q dµ B puq.
Finally since 1{C ď r ď C, the right hand side tends to zero as n Ñ 8. Hence the integral of logpr ˝f ˚{rq vanishes, and (5.11) follows from Furstenberg's formula.

Proposition 5.10. The point Ppepωqq is ν N -almost surely extremal in PpKahpXqq and in PpPsefpXqq.

Proof. The class epωq almost surely belongs to KahpXq and to the isotropic cone. By the Hodge index theorem -more precisely, by the case of equality in the reverse Schwarz Inequality (2.6)epωq cannot be a non-trivial convex combination of classes with non-negative intersection and mass 1; so Ppepωqq is an extremal point of the convex set PpKahpXqq Ă PH 1,1 pX; Rq. From Proposition 2.3, there are at most countably many points Ppuq in PpKahpXqq such that u 2 " 0 and Ppuq is not extremal in PpPsefpXqq. Therefore the second assertion follows from the fact that µ B is atomless. 5.4. Some estimates for random products of matrices. The aim of this section is to establish some technical facts which will play a crucial role in our study of the closed positive currents T s ω in Section 6. The key results are Theorem 5.11 and Lemma 5.13.

5.4.1.

Sequences of good times. We now describe a theorem of Gouëzel and Karlsson, specialized to our specific context. Fix a base point e 0 in the hyperbolic space H X , for instance e 0 " rκ 0 s with κ 0 a fixed Kähler form (as in Section 2.2). Consider the two functions of pn, ωq P N ˆΩ defined by (5.12) T pn, ωq " d H pe 0 , pf n ω q ˚e0 q, N pn, ωq " log }pf n ω q ˚}. They satisfy the subadditive cocycle property (5.13) apn `m, ωq ď apn, ωq `apm, σ n pωqq, where σ is the unilateral shift on Ω (see § 4.3). Let apn, ωq be such a subadditive cocycle; if ap1, ωq is integrable the asymptotic average is defined to be the limit (5.14) A " lim nÑ`8

1 n ż apn, ωq dν N pωq;

it exists in r´8, `8q, and we say it is finite if A ‰ ´8. The functions T and N are examples of ergodic subadditive cocycles and from Theorem 5.7, Remark 5.9, and Corollary 2.5, we deduce that the asymptotic average of each of these cocycles is equal to λ H 1,1 .

Following [START_REF] Gouëzel | Subadditive and multiplicative ergodic theorems[END_REF], we say that apn, ωq is tight along the sequence of positive integers pn i q if there is a sequence of real numbers pδ q " pδ pωqq ě0 such that (i) δ converges to 0 as goes to `8; (ii) for every i, and for every 0 ď ď n i , ˇˇapn i , ωq ´apn i ´ , σ pωqq ´A ˇˇď δ ;

(iii) for every i and for every 0 ď ď n i apn i , ωq ´apn i ´ , ωq ě pA ´δ q .

Theorem 5.11 (Gouëzel and Karlsson [START_REF] Gouëzel | Subadditive and multiplicative ergodic theorems[END_REF]). Let apn, ωq be an ergodic subadditive cocycle, with a finite asymptotic average A. Then, for almost every ω, the cocycle is tight along a subsequence pn i pωqq of positive upper density.

Recall that the (asymptotic) upper density of a subset S of N is the non-negative number defined by denspSq " lim sup kÑ`8 `1 k |S X r0, k ´1s| ˘. A sequence pn i q iě0 is said to have positive upper density if the set of its values S " tn i ; i ě 0u satisfies denspSq ą 0.

Proof. Let us explain how this result follows from [START_REF] Gouëzel | Subadditive and multiplicative ergodic theorems[END_REF]. First, fix a small positive real number ρ ą 0, and apply Theorem 1.1 and Remark 1.2 of [START_REF] Gouëzel | Subadditive and multiplicative ergodic theorems[END_REF] to get a set Ω ρ of measure 1 ´ρ such that the first two properties (i) and (ii) are satisfied for every ω P Ω ρ with respect to a sequence pδ q that does not depend on ω, and for a sequence of times pn i pωqq of upper density ě 1 ´ρ. To get (iii), we apply Lemma 2.3 of [START_REF] Gouëzel | Subadditive and multiplicative ergodic theorems[END_REF] to the sub-additive cocycle apn, ωq (not to the cocycle bpn, ωq " apn, σ ´npωqq as done in [START_REF] Gouëzel | Subadditive and multiplicative ergodic theorems[END_REF]). For every ε ą 0, there is a subset Ω 1 ε Ă Ω and a sequence pδ 1 q ě0 such that (a) ν N pΩ 1 ε q ą 1 ´ε, and δ 1 converges towards 0 as goes to `8; (b) for every ω P Ω 1 ε , there is a set of bad times Bpωq Ă N such that for every k ě 0 |Bpωq X r0, k ´1s| ď εk, and for every n R Bpωq and every 0 ď ď n, apn, ωq ´apn ´ , ωq ě pA ´δ1 q .

If ω belongs to Ω ρ X Ω 1 ε , the set of indices i for which n i pωq R Bpωq is infinite. More precisely, the set Spωq " tn j pωq ; n j pωq R Bpωqu has asymptotic upper density ě 1 ´ρ ´ε. Along this subsequence, the three properties (i), (ii), and (iii) are satisfied. Since this holds for all ω P Ω 1 ε X Ω ρ and the measure of this set is ě 1 ´ρ ´ε, this holds for ν N -almost every ω. Corollary 5.12. For ν N -almost every ω P AutpXq N , there is an increasing sequence of integers pn i pωqq going to `8 and a real number Apωq such that

n i pωq ÿ j"0 › › `f j ω ˘˚› › › › `f n i pωq ω ˘˚› ›
ď Apωq and

n i pωq ÿ j"0 › › `f n i pωq´j σ j pωq ˘˚› › › › `f n i pωq ω ˘˚› › ď Apωq
for all indices i ě 0.

Proof. Apply Theorem 5.11 to the subadditive cocyle N pn, ωq and note that (5.15)

n i pωq ÿ j"0 › › `f j ω ˘˚› › › › `f n i pωq ω ˘˚› › " n i pωq ÿ "0 › › `f n i ´ ω ˘˚› › › › `f n i ω ˘˚› › " n i pωq ÿ "0
e N pn i ´ ,ωq e N pn i ,ωq ď

n i pωq ÿ "0
e

´ pλ H 1,1 ´δ q which is bounded as n i pωq Ñ 8. The second estimate is similar.

5.4.2.

A mass estimate for pull-backs. Assume that pX, νq is non-elementary and satisfies the condition (4.1). Recall from Lemma 5.5 that Mppf n ω q ˚aq ´1pf n ω q ˚a converges to the pseudoeffective class epωq for almost every ω and every Kähler class a. Thus, on a set of total ν Nmeasure, this convergence holds for all σ k pωq, k ě 0. Since Mpepωqq " 1, we obtain (5.16) f 0 epσωq " Mpf 0 epσωqqepωq; more generally, for every k ě 1,

(5.17) pf k ω q ˚epσ k ωq " Mppf k ω q ˚epσ k ωqqepωq. Lemma 5.13. For ν N -almost every ω, we have

1 n log Mppf n ω q ˚epσ n ωqq ÝÑ nÑ8 λ H 1,1 .
This does not follow from Lemma 5.3 because epσ n ωq depends on n.

Proof. For almost every ω, for every k ě 1, and for every Kähler class a, epσ k ωq is the limit of Mpf k ¨¨¨f n´1 aq ´1f k ¨¨¨f n´1 a as n goes to `8. So (5.18)

f 0 ¨¨¨f k´1 epσ k pωqq " ˆlim nÑ8 Mpf 0 ¨¨¨f n´1 aq Mpf k ¨¨¨f n´1 aq ˙epωq ": ζpk, ωqepωq
where ζpk, ωq is both equal to Mppf k ω q ˚epσ k pωqqq and to the limit

(5.19) ζpk, ωq " lim nÑ8 Mpf 0 ¨¨¨f n´1 aq Mpf k ¨¨¨f n´1 aq " lim nÑ8 Mppf n ω q ˚aq Mppf n´k σ k pωq q ˚aq .
We want to show that, ν N -almost surely, p1{kq log ζpk, ωq converges to λ H 1,1 .

Before starting the proof, note that ζ is a multiplicative cocycle: ζpk, ωq " ś k "1 ζp1, σ ωq; in particular, log ζpk, ωq is equal to the Birkhoff sum ř k "1 log ζp1, σ ωq. Since (5.20)

C ´1› › pf ´1 0 q ˚› › H 1,1 ď Mpf 0 epσpωqqq ď C}f 0 } H 1,1
, our moment condition shows that logpζp1, ωqq is integrable. So, by the ergodic theorem of Birkhoff, lim k 1 k log ζpk, ωq exists ν N -almost surely. Pick a sequence pn i q of good times for ω, as in Theorem 5.11. If we compute the limit in Equation (5.19) along the subsequence pn i q we see that ζpk, ωq ě C expppλ H 1,1 ´δpkqqkq for some constant C ą 0, and some sequence δpkq converging to 0 as k goes to `8. This gives (5.21) lim sup kÑ`8

1 k log ζpk, ωq ě λ H 1,1 .
Now, consider the linear cocycle Υ : Ω ˆH1,1 pX, Rq Ñ Ω ˆH1,1 pX, Rq defined by (5.22) Υpω, uq " pσpωq, pf 1 ω q ˚uq and let PΥ be the associated projective cocycle on Ω ˆPH 1,1 pX, Rq. The Lyapunov exponents of Υ are ˘λH 1,1 , each with multiplicity 1, and 0, with multiplicity h 1,1 pXq ´2. Since Pppf 1 ω q ˚epσpωqqq " Ppepωqq, the measurable section tpω, Ppepωqqq ; ω P Ωu is PΥ-invariant. Therefore, by ergodicity of σ with respect to ν N , m " ş δ Ppepωqq dν N pωq defines an invariant and ergodic measure for PΥ. It follows from the invariance of the decomposition into characteristic subspaces in Oseledets' theorem that epωq is contained in a given characteristic subspace of the cocycle Υ; thus, if λ denotes the Lyapunov exponent of Υ in that characteristic subspace, we get (as in Remark 5.9) that λ " Ledrappier [86,§1.5]). Birkhoff's ergodic theorem implies that lim k 1 k log ζpk, ωq " ´λ, with λ P t˘λ H 1,1 , 0u, therefore the Inequality (5.21) concludes the proof. 5.4.3. Exponential moments. The result of this section will only be used in Theorem 6.16 so this paragraph may be skipped on a first reading. Consider the exponential moment condition (5.23) Dτ ą 0,

ż log ˇˇpf 1 ω q ˚u| u| dmpω, uq " ż log Mppf 1 ω q ˚pepωqq Mpepωqq dν N pωq " ż log ζp1, ωq ´1 dν N pωq (see
ż `}f } C 1 `› › f ´1› › C 1 ˘τ dνpf q ă `8.
As in Section 5.1, this upper bound implies the cohomological moment condition

(5.24) Dτ ą 0, ż `}f ˚}H 1,1 `› › pf ´1q ˚› › H 1,1 ˘τ dνpf q ă `8.
Proposition 5.14. Assume that ν satisfies the Condition (5.23). Let D : AutpXq Ñ R `be a measurable function such that ş Dpf q τ 1 dνpf q ă 8 for some τ 1 ą 0. Then, there is a measurable function B : Ω Ñ R `satisfying ż log `pBpωqq dν N pωq ă 8

such that for ν N -almost every ω " pf n q and every n ě 0

n´1 ÿ j"1 Dpf j´1 q › › f j ¨¨¨f n´1 › › › › f 0 ¨¨¨f n´1 › › ď Bpωq, and n´1 ÿ j"1 Dpf j q › › f 0 ¨¨¨f j´1 › › › › f 0 ¨¨¨f n´1 › › ď Bpωq.
This is a refined version of Corollary 5.12.

Proof. We are grateful to Sébastien Gouëzel for explaining this argument to us. We temporarily use the notation Pp¨q for probability with respect to ν n or ν N (so, here, P does not denote projectivisation).

First Estimate.-We start with the first estimate:

ř n´1 j"1 Dpf j´1 q › › f j ¨¨¨f n´1 › › }f 0 ¨¨¨f n´1 } ď Bpωq.
Step 1.-For every 0 ă ε ă λ H 1,1 there exists constants c, C ą 0 such that (5.25) P p|pf n ω q ˚b| ď e εn q ď Ce ´cn . for every b P Π Γ with |b| " 1. This large deviation result, which is uniform in n and b, follows from condition (5.24) (see for instance [18, §V.6], and [12, §12]).

Step 2.-Let us prove that (5.26)

P ˜› › f j ¨¨¨f n´1 › › › › f 0 ¨¨¨f n´1 › › ą e ´εj ¸ď Ce ´cj .
For this, fix f j , . . . , f n´1 . Then, there is a point a P Π Γ with |a| " 1 such that

› › ›f j ¨¨¨f n´1 › › › " ˇˇf j ¨¨¨f n´1 a ˇˇ. Hence, if › › f 0 ¨¨¨f n´1 › › ă › › f j ¨¨¨f n´1
› › e εj , we infer that

(5.27) ˇˇf 0 ¨¨¨f n´1 a ˇˇă › › f j ¨¨¨f n´1
› › e εj " ˇˇf j ¨¨¨f n´1 a ˇˇe εj . Thus, if we set (5.28) b "

1 ˇˇf j ¨¨¨f n´1 a ˇˇf j ¨¨¨f n´1 a,
we obtain that ˇˇf 0 ¨¨¨f j´1 b ˇˇă e εj ; this happens with (conditional) probability ď Ce ´cj (relative to ν ˚j ), for the uniform constants given in Step 1. Averaging over f j , . . . , f n´1 , we get the result.

Step 3.-The moment condition satisfied by D and Markov's inequality imply PpD ą Kq ď C 1 K ´τ 1 for some constant C 1 ą 0. Fix ε P R ˚small with respect to λ H 1,1 and τ 1 . Then, on a set Ωpε, Jq of measure

(5.29) ν N pΩpε, Jqq ě 1 ´C2 pe ´pετ 1 {2qJ `e´εcJ q,
for some C 2 " C 2 pεq ą 0, we have both Dpf j´1 q ď e εj{2 and }f j ¨¨¨f n´1 } }f 0 ¨¨¨f n´1 } ď e ´εj for all j ě J.

For ω " pf n q in Ωpε, Jq, we get

n´1 ÿ j"1 Dpf j´1 q › › f j ¨¨¨f n´1 › › › › f 0 ¨¨¨f n´1 › › ď J ÿ j"1 Dpf j´1 q › › f j ¨¨¨f n´1 › › › › f 0 ¨¨¨f n´1 › › `n´1 ÿ j"J`1 e ´εj{2 (5.30) ď J ÿ j"1 Dpf j´1 q › › ›pf ´1 j´1 q ˚¨¨¨pf ´1 0 q ˚› › › `C3 " C 3 `J´1 ÿ j"0 }f 0 } ¨¨¨› › f j › › Dpf j q.
The moment condition (5.23) gives Pp}f ˚} ą Kq ď C 4 K ´τ and as already noticed, we also have PpDpf q ą Kq ď C 1 K ´τ 1 . So, with η " minpτ, τ 1 q, there is a set of probability at least 1 ´C5 JK ´η on which (5.31)

J´1 ÿ j"0 Dpf j q}f 0 } ¨¨¨› › f j › › ď C 6 JK J`2 .
Taking K " J 3{η , we have JK ´η " J ´2, and we obtain (5.32)

P ˜J´1 ÿ j"0 Dpf j q}f 0 } ¨¨¨› › f j › › ą J 1`3pJ`2q{η ¸ď C 7 J ´2.
Also, note that J 1`p3J`6q{η ď exp `CJ 3{2 ˘.

By the Borel-Cantelli lemma, the sum in (5.30) is almost surely bounded by some constant Bpωq which satisfies P `log B ą J 3{2 ˘ď CJ ´2; in particular E `log `B˘ă

8.

Second Estimate.-To obtain the second estimate of Proposition 5.14, we apply the above proof to the reversed random dynamical system, induced by ν : f Þ Ñ νpf ´1q. Indeed, the core of the argument is the inequality (5.30) which is not sensitive to the order of compositions.

LIMIT CURRENTS

Our goal in this section is to prove the counterpart of the convergence (5.7) at the level of closed positive currents on X. Throughout this section we fix a non-elementary random holomorphic dynamical system pX, νq satisfying the moment condition (4.1), so that all results of §5 apply. We refer the reader to [START_REF] Guedj | Degenerate complex Monge-Ampère equations[END_REF] (in particular Chapter 8) for basics on pluripotential theory on compact Kähler manifolds (see also [START_REF] Demailly | Analytic methods in algebraic geometry[END_REF]).

6.1. Potentials and cohomology classes of positive closed currents. Let us fix once and for all a family of Kähler forms pκ i q 1ďiďh 1,1 pXq such that rκ i s 2 " 1 and the rκ i s form a basis of H 1,1 pX; Rq; in addition we require that the κ i satisfy (6.1)

κ 0 " β ÿ i κ i
for some β ą 0, where κ 0 is the Kähler form chosen in Section 2.2 (note that necessarily β ă 1). We also fix a smooth volume form vol X on X, normalized by ş X vol " 1. On tori, K3 and Enriques surfaces, we choose vol X to be the canonical AutpXq-invariant volume form (see Remark 3.15). It is convenient to assume in all cases that vol X is also the volume form associated with the Kähler metric κ 0 (up to scaling).

Unless otherwise specified, the currents we shall consider will be of type p1, 1q. The action of a current T on a test form ϕ will be denoted by xT, ϕy or ş T ^ϕ. If T is closed, we denote its cohomology class by rT s; so, if ϕ is a closed form, xT, ϕy " xrT s | rϕsy. By definition the mass of a current is the quantity MpT q " ş T ^κ0 ; so MpT q " xrT s|rκ 0 sy when T is closed.

6.1.1. Normalized potentials. If a is an element of H 1,1 pX; Rq, we denote by pc i paqq 1ďiďh 1,1 pXq its coordinates in the basis prκ i sq, so that a " ř i c i paqrκ i s. Then, we set (6.2)

Θpaq "

ÿ i c i paqκ i .
Likewise, given a closed p1, 1q-form α or a closed current of bidegree p1, 1q, we set c i pαq " c i prαsq and Θpαq " Θprαsq; hence, rΘpαqs " rαs. It is worth keeping in mind that some coefficients c i pαq can be negative and Θpαq need not be semi-positive, even if α is a Kähler form. If T is a closed positive current of bidegree p1, 1q on X we define its normalized potential to be the unique function u T P L 1 pXq such that (6.3) T " ΘpT q `dd c pu T q and ż X u T vol " 0

(see [73, §8.1]). The function u T is locally given as the difference v ´w of a psh potential v of T and a smooth potential w of ΘpT q.

Lemma 6.1. There is a constant A ą 0 such that the following properties are satisfied for every closed positive current T of mass 1

(1) ´A ď c i pT q ď A for all 1 ď i ď h 1,1 pXq, and ´Aκ 0 ď ΘpT q ď Aκ 0 .

(2) the function u T is pAκ 0 q-psh: dd c pu T q `Aκ 0 is a positive current.

Proof. Since the coefficients T Þ Ñ c i pT q are continuous functions on the space of currents and closed positive currents of mass 1 form a compact set K, the functions |c i | are bounded by some uniform constant A 1 on K. Setting A " A 1 β ´1, with β as in Equation (6.1), we get ´Aκ 0 ď ΘpT q ď Aκ 0 for all T P K. Then dd c u T " T ´ΘpT q ě ´Aκ 0 and (2) follows.

Corollary 6.2. The set of potentials tu T | T is a closed positive current of mass 1 on Xu is a compact subset of L 1 pX; volq.

Proof. Since this is a set of pAκ 0 q-psh functions which are normalized with respect to a smooth volume form, the result follows from Proposition 8.5 and Remark 8.6 in [START_REF] Guedj | Degenerate complex Monge-Ampère equations[END_REF].

Remark 6.3. Another usual normalization imposes the condition sup xPX u T pxq " 0; by compactness this would only change u T by some uniformly bounded constant. However since many of our dynamical examples preserve a natural volume form it is more convenient for us to normalize as in (6.3).

6.1.2. The diameter of a pseudo-effective class. For a class a P PsefpXq we define (6.4) Curpaq " tT ; T is a closed positive current with rT s " au, This is a compact convex subset of the space of currents. If S and T are two elements of Curpaq, then ΘpSq " ΘpT q " Θpaq and T ´S " dd c pu T ´uS q. We set

(6.5) distpS, T q " ż X |u S ´uT | vol .
This is a distance that metrizes the weak topology on Curpaq: this follows for instance from the fact that by Corollary 6.2 pCurpaq, distq is compact. By definition, the diameter of a is (6.6) Diampaq " DiampCurpaqq " suptdistpS, T q ; S, T in Curpaqu,

If a P PsefpXq, then Diampaq is a non-negative real number which is finite by Corollary 6.2.

If Curpaq " H, we set Diampaq " ´8. Note that Diam is homogeneous of degree 1: Diamptaq " t Diampaq for every a P PsefpXq and t ą 0.

Lemma 6.4. On PsefpXq, a Þ Ñ Diampaq is upper semi-continuous, hence measurable.

Proof. Let pa n q be a sequence of pseudo-effective classes converging to a. For every n we choose a pair of currents pS n , T n q in Curpa n q 2 such that distpS n , T n q ě Diampa n q ´1{n. The masses of S n and T n are uniformly bounded because they depend only on a n . By Corollary 6.2, we can extract a subsequence such that S n and T n converge towards closed positive currents S, T P Curpaq, and u Sn and u Tn converge towards their respective potentials u S and u T in L 1 pX, volq. Then, distpS, T q " ş X |u S ´uT |vol " lim n distpS n , T n q, which shows that Diampaq ě lim sup n pDiampa n qq.

6.2. Action of AutpXq.

6.2.1.

A volume estimate. Let X be a compact, complex manifold, and let vol be a C 0 -volume form on X with volpXq " 1. If f is an automorphism of X, let Jacpf q : X Ñ R denote its Jacobian determinant with respect to the volume form vol: f ˚vol " Jacpf qvol. The following lemma is a variation on well-known ideas in holomorphic dynamics (see for instance [START_REF] Guedj | Decay of volumes under iteration of meromorphic mappings[END_REF]). Lemma 6.5. Let κ be a hermitian form on X. Let h be a κ-psh function on X such that ş X h vol " 0, and let f be an automorphism of X. Then, ż

X |h ˝f | vol ď C logpC › › Jacpf ´1q › › 8 q
for some positive constant C that depends on pX, κq but neither on f nor on h.

Proof. We first observe that there is a constant c ą 0 such that volt|h| ě tu ď c expp´t{cq; this follows from Lemma 8.10 and Theorem 8.11 in [START_REF] Guedj | Degenerate complex Monge-Ampère equations[END_REF], together with Chebychev's inequality (see Remark 6.3 for the normalization). Then, we get

ż X |h ˝f | vol " ż 8 0 volt|h ˝f | ě tudt (6.7) " ż 8 0 volpf ´1t|h| ě tuqdt ď ż s 0 volpXqdt `› › Jacpf ´1q › › 8 ż 8 s c expp´t{cqdt ď s volpXq `› › Jacpf ´1q › › 8 c 2 expp´s{cq (6.8)
where the inequality in the third line follows from the change of variable formula. Now, we minimize (6.8) by choosing s " c logpc › › Jacpf ´1q › › 8 {volpXqq and we infer that (6.9)

ż X |h ˝f | vol ď c volpXq ˜1 `log ˜c› › Jacpf ´1q › › 8 volpXq ¸¸.
Since the total volume is invariant, }Jacpf q} 8 ě 1, and the asserted estimate follows.

6.2.2. Equivariance. Let us come back to the study of pX, νq. If f is an automorphism of X, then f ˚Curpaq " Curpf ˚paqq for every class a P H 1,1 pX, Rq. If a P PsefpXq and T P Curpaq, then T " Θpaq `dd c pu T q and (6.10) f ˚T " f ˚Θpaq `dd c pu T ˝f q " Θpf ˚aq `dd c pu f ˚Θpaq `uT ˝f q.

This shows that the normalized potential of f ˚T is given by (6.11)

u f ˚T " u f ˚Θpaq `uT ˝f `Epf, T q
where Epf, T q P R is the constant for which the integral of u f ˚T vanishes; since u f ˚Θpaq has mean 0, we get (6.12) Epf, T q " ´żX `uf ˚Θpaq `uT ˝f ˘vol " ´żX u T ˝f vol.

Remark 6.6. If vol is f -invariant, for instance if it is the canonical volume on a K3 or Enriques surface, then Epf, T q " 0, which simplifies a little bit the analysis of the potentials below.

Lemma 6.7. On the set of closed positive currents of mass 1, the function pf,

T q Þ Ñ Epf, T q satisfies |Epf, T q| ď C log `C› › Jacpf ´1q › › 8
where the implied positive constant C depends neither on f nor on T .

Proof. From Lemma 6.1, the potentials u T are uniformly pAκ 0 q-psh, so the conclusion follows from Equation (6.12) and Lemma 6.5.

Lemma 6.8. There exists a constant C such that if a is any pseudo-effective class of mass 1, and f is any automorphism of X, then

Diampf ˚aq ď C log `C› › Jacpf ´1q › › 8 ˘.
Proof. Indeed, if S and T belong to Curpaq, by Equation (6.11) we have u f ˚T ´uf ˚S " pu T ´uS q ˝f `Epf, T q ´Epf, Sq, so

(6.13) distpf ˚T , f ˚Sq ď ż |u T ˝f | vol `ż |u S ˝f | vol `|Epf, T q| `|Epf, Sq| ;
and the result follows from Lemmas 6.5 and 6.7, since u S and u T are uniformly pAκ 0 q-psh.

6.2.3. An estimate for canonical potentials.

Lemma 6.9. For any Kähler form κ on X there exists a positive constant Cpκq such that for every f P AutpXq,

› › u f ˚κ› › C 1 ď Cpκq}f } 2 C 1 .
In addition Cpκq ď C 1 }κ} 8 , where }κ} 8 is the sup norm of the coefficients of κ in a system of coordinate charts, and C 1 depends only on X (and the choice of these coordinate charts).

Recall the choice of Kähler forms pκ i q from § 6.1 and the definition of Θp¨q from § 6.1.1. Corollary 6.10. If κ " ř i c i κ i in Lemma 6.9, then the constant Cpκq satisfies Cpκq ď

C 2 Mpκq. Likewise, › › u f ˚Θpaq › › C 1 ď C 3 Mpaq}f } 2
C 1 for all a P PsefpXq.

Indeed Cpκq ď C 1 }κ} 8 ď C 2 ř i |c i | and u f ˚Θpaq " ř c i paqu f ˚κi .
Proof of Lemma 6.9. By definition f ˚κ ´Θpf ˚κq " dd c `uf ˚κ˘. The desired estimate will be obtained by constructing a solution φ to the equation (6.14)

dd c φ " f ˚κ ´Θpf ˚κq which satisfies }φ} C 1 ď C}f } 2 C 1 .
Then, since u f ˚κ and φ differ by a constant and u f ˚κ is known to vanish at some point, it follows that u f ˚κ satisfies the same estimate. To construct the potential φ, we follow the method of Dinh and Sibony [50, Prop. 2.1] which is itself based on [START_REF] Bost | Heights of projective varieties and positive Green forms[END_REF] (we keep the notation from [START_REF] Dinh | Green currents for holomorphic automorphisms of compact Kähler manifolds[END_REF]). Let α be a closed p2, 2q-form on X ˆX which is cohomologous to the diagonal ∆. In [START_REF] Bost | Heights of projective varieties and positive Green forms[END_REF], Bost, Gillet and Soulé construct an explicit p1, 1qform K on X ˆX such that dd c K " r∆s ´α; they refer to it as the "Green current". It is C 8 outside the diagonal, and along ∆, it satisfies the estimates (here we mean that these estimates hold for the coefficients of K and ∇K in local coordinates). These estimates are easily deduced from the explicit expression of K as π ˚p p ϕη ´βq given in the proof of Proposition 2.1 of [START_REF] Dinh | Green currents for holomorphic automorphisms of compact Kähler manifolds[END_REF], where π : { X ˆX Ñ X ˆX is the blow-up of the diagonal, η and β are smooth (1,1) forms on { X ˆX and p ϕ is a function with logarithmic singularities along the proper transform of ∆ in X ˆX. It is shown in [50, (for some constant C 2 depending only on X). A similar estimate for ∇φ is obtained from derivation under the integral sign and the fact that ∇K P L p loc for p ă 4{3. This concludes the proof.

6.3. Convergence and extremality. Theorem 6.11. Let pX, νq be a non-elementary random holomorphic dynamical system on a compact Kähler surface X, satisfying the moment condition (4.1). Then for µ B -almost every point a P LimpΓq, the following properties hold:

(1) there is a unique nef and isotropic class a P H 1,1 pX; Rq of mass 1 with Ppaq " a;

(2) the convex set Curpaq is a singleton tT a u;

(3) the class a is an extremal point of PpKahpXqq and of PpPsefpXqq; (4) the current T a is extremal in the convex set of closed positive currents of mass 1.

Combining this result with Lemma 5.6 and Equation (5.9) we obtain the first and second assertions of the following corollary; the third assertion follows from the first one and the equivariance relation (5.16).

Corollary 6.12. The following properties are satisfied for ν N -almost every ω:

(1) there exists a unique closed positive current T s ω in the cohomology class epωq; (2) for every Kähler form κ, 1 M ppf n ω q ˚κq pf n ω q ˚κ ÝÑ nÑ8 T s ω .

(3) the currents T s ω satisfy the equivariance property pf ω q ˚T s σpωq "

Mppf ω q ˚T s σpωq q MpT s ω q T s ω " Mppf ω q ˚T s σpωq qT s ω .

Proof of Theorem 6.11. The first and third properties were already established, respectively in Lemma 2.15 and 2.16 and Proposition 5.10. Property (4) follows from ( 2) and (3). It remains to prove [START_REF] Bahnmüller | A Margulis-Ruelle inequality for random dynamical systems[END_REF]. For this, we denote by f ˚the projective action of f ˚on PH

`C› › Jacpf 1 ˝¨¨¨˝f n q ´1› › 8 M pf n ¨¨¨f 1 aq ď C ř n´1 i"0 log › › f ´1 i › › C 1
Mpf n ¨¨¨f 1 aq .

We conclude with two remarks. Firstly, the moment condition (4.1) implies that the sequence

1 n ř n´1 i"0 log › › f ´1 i › › C 1 is almost surely bounded.
Secondly, Lemma 5.5 shows that Mpf n ¨¨¨f 1 aq goes exponentially fast to infinity for ν N -almost every ω " pf n q (this is where the order of compositions matters). Thus diam `f n ¨¨¨f 1 paq ˘Ñ 0 almost surely, and we are done. Remark 6.13. The uniqueness of T a in its cohomology class implies that T a depends measurably on a. Indeed there is a set E Ă LimpΓq of full measure along which the map a Þ Ñ T a is continuous (recall that the space Cur 1 pXq of positive closed currents of mass 1 on X is a compact metrizable space). This implies that a Þ Ñ T a is a measurable map from LimpΓq, endowed with the µ B -completion of the Borel σ-algebra, to Cur 1 pXq, endowed with its Borel σ-algebra.

6.4. Continuous potentials. We now study the limit currents T s ω introduced in Corollary 6.12. Theorem 6.14. Let pX, νq be a non-elementary random holomorphic dynamical system on a compact Kähler surface X, satisfying the moment condition (4.1). Then for ν N -a.e. ω the current T s ω has continuous potentials. Lemma 6.15. Let κ be any Kähler form on X. For ν N -almost every ω, there exists an increasing sequence of integers pn i q iě0 " pn i pωqq such that (1) the potentials of the pull-back currents Mppf n i ω q ˚κq ´1u pf n i ω q ˚κ are uniformly bounded;

(2) the same holds for the push-forward currents Mppf n i ω q ˚κq ´1u pf n i ω q˚κ . If the exponential moment condition (5.23) holds, these assertions hold for all n (i.e. extracting a subsequence pn i q is not necessary); in addition the function

ω Þ Ñ log `› › u T s ω › › 8 is ν N -integrable.
Proof of the Lemma. Recall the notation ω " pf n q ně0 . First,

f n´1 κ " f n´1 Θpκq `dd c pu κ ˝fn´1 q (6.22)
" Θpf n´1 κq `dd c ´uf n´1 Θpκq `uκ ˝fn´1 ¯ (For the moment, we do not introduce the constants Epf n ; κq in the computation). We obtain

f n´2 f n´1 κ " f n´2 Θpf n´1 κq `dd c ´uf n´1 Θpκq ˝fn´2 `uκ ˝pf n´1 ˝fn´2 q " Θpf n´2 f n´1 κq `dd c ´uf n´2 Θpf n´1 κq `uf n´1 Θpκq ˝fn´2 `uκ ˝pf n´1 ˝fn´2 q ¯.
Setting G j,k " f k´1 ˝¨¨¨˝f j , for j ď k ´1, (so in particular G 0,j " f j ω for all j ě 1) and G j,j " id X , we get

pf n ω q ˚κ " Θppf n ω q ˚κq `dd c ˜uκ ˝f n ω `n´1 ÿ j"0 u f j ΘpG j`1,n κq ˝G0,j ¸. (6.23)
Let u n denote the function in the parenthesis. We want to estimate the sup-norm }u n } 8 . Lemma 6.9 and Corollary 6.10 provide successively the following upper bounds

› › ›u f j ΘpG j`1,n κq › › › 8 ď C}f j } 2 C 1 MpG j`1,n κq ď CMpκq}f j } 2 C 1 › › G j`1,n › › , (6.24) (6.25) › › › › 1 Mppf n ω q ˚κq u n › › › › 8 ď }u κ } 8 Mppf n ω q ˚κq `CMpκq n´1 ÿ j"0 }f j } 2 C 1 › › ›G j`1,n › › › Mppf n ω q ˚κq .
To estimate this sum we apply Theorem 5.11 to the subadditive cocycle N pn, ωq " log }pf n ω q ˚}, as we did for Corollary 5.12: there exists a sequence pδ j q of positive numbers converging to 0, an increasing sequence n i " n i pωq of integers, and a constant C 1 pωq such that (6.26)

› › G j`1,n i › › Mppf n i ω q ˚κq - › › f j`1 ¨¨¨f ni ´1› › › › f 0 ¨¨¨f ni ´1› › ď C 1 expp´pλ 1 ´δj
qjq for all i ě 1 and all 0 ď j ď n i . Fix any real number ε with 0 ă ε ă λ 1 . Then from Lemma 4.3, we know that, for almost every ω, there is a constant C 2 pωq such that }f j } 2 C 1 ď C 2 exppεjq. So from (6.25) we get (6.27)

› › › › 1 Mppf n i ω q ˚κq u n i › › › › 8 ď }u κ } 8 Mppf n i ω q ˚κq `C3 pωqMpκq n i ´1 ÿ j"0 expp´pλ 1 ´ε ´δpjqqjq This inequality shows that › › Mppf n i ω q ˚κq ´1u n i › › 8 is uniformly bounded. Now, note that u pf n ω q ˚κ " u n `En with E n " ´ş u n vol. Since › › Mppf n i ω q ˚κq ´1u n i › ›
8 is uniformly bounded, so is Mppf n i ω q ˚κq ´1E n i , and the first assertion of the lemma is established. The second assertion is proved exactly in the same way, except that the expressions of the form f j ΘpG j`1,n κq must be replaced by pf ´1 n´j q ˚Θppf ´1 0 ˝¨¨¨˝f ´1 n´j´1 q ˚κq; then we use the second estimate in Corollary 5.12, and the fact that for every f P AutpXq, }f ˚} -› › pf ´1q ˚› › . If the exponential moment condition (5.23) holds, we follow the same argument and apply Proposition 5.14 -instead of Theorem 5.11 -to (6.25), with Dpf q " }f } 2 C 1 .

Proof of Theorem 6.14. First, we prove that the normalized potential u T s ω is bounded, for ν Nalmost every ω. To see this, recall that Mppf n ω q ˚κq ´1pf n ω q ˚κ converges to T s ω as n Ñ 8. From Lemma 6.15, we know that the normalized potentials Mppf n ω q ˚κq ´1u pf n ω q ˚κ of the currents Mppf n ω q ˚κq ´1 pf n ω q ˚κ are uniformly bounded along some subsequence n i " n i pωq. These potentials are Aκ 0 -psh functions on X so, by compactness, they converge to u T s ω in L 1 pX; volq.

Thus, u T s ω is essentially bounded. We conclude that u T s ω is bounded because quasi-plurisubharmonic functions are upper semi-continuous and have a value (in R Y t´8u) at every point. Now, we show that u T s ω is continuous. Here, the argument is similar to the one used to prove Theorem 6.11. If T is a positive closed current with bounded potential on X, we define (6.28)

JumppT q " max xPX ˆlim sup

yÑx u T pyq ´lim inf yÑx u T pyq ˙.
Then 0 ď JumppT q ď 2}u T } 8 , and JumppT q " 0 if and only if u T is continuous. In addition Jumppf ˚T q " JumppT q for every f P AutpXq because f ˚T " Θpf ˚aq `dd c pu f ˚Θpaq `uT f q and u f ˚ΘprT sq is continuous (see Equation (6.10)). From the equivariance relation (6.29)

T s ω " 1 M `pf n ω q ˚T s σ n ω ˘T s σ n ω ,
which follows from the third assertion of Corollary 6.12, we get (6.30)

Jump pT s ω q " 1 M `pf n ω q ˚T s σ n ω ˘Jump pT s σ n ω q . Remark 6.13 says that ω Þ Ñ T s ω is measurable; hence, ω Þ Ñ u T s ω is measurable. If C is large enough, the first step of the proof gives a subset Ω C Ă Ω such that νpΩ C q ą 0 and › › u T s ω › › 8 ď
C for all ω P Ω C . By ergodicity of the shift, σ n ω P Ω C for almost every ω and infinitely many n; for such an n, › › u T s σ n ω › › 8 ď C and Jump pT s σ n ω q ď 2C. By Lemma 5.13, M `pf n ω q ˚T s σ n ω ˘goes to infinity almost surely. So, Jump pT s ω q " 0, and the proof is complete. Theorem 6.16. Let pX, νq be a non-elementary random holomorphic dynamical system on a compact Kähler surface X, satisfying the exponential moment condition (5.23). Then there exists θ ą 0 such that for ν N -almost every ω the potential u T s ω is Hölder continuous of exponent θ.

Proof. The initial computations are similar (but not identical) to those used to reach Lemma 6.15.

Keeping the notation G j,n " f n´1 ˝¨¨¨˝f j , a descending induction starting from (6.31)

f n´1 T s σ n ω " Θpf n´1 T s σ n ω q `dd c ´uf n´1 ΘpT s σω q `uT s σ n ω ˝fn´1 ȳields pf n ω q ˚T s σ n ω " Θ ppf n ω q ˚T s σ n ω q `dd c ˜n´1 ÿ j"0 u f j ΘpG j`1,n T s σ n ω q ˝f j ω `uT s σ n ω ˝f n ω ¸. (6.32)
Thus, there is a constant of normalization E " Epω; nq such that (6.33)

u T s ω " 1 Mppf n ω q ˚pT s σ n ω qq ˜n´1 ÿ j"0 u f j ΘpG j`1,n T s σ n ω q ˝f j ω `uT s σ n ω ˝f n ω ¸`E.
Note that the additional term E does not affect the modulus of continuity of u T s ω . Since Lippf j q ď }f j } C 1 for all j, Lemma 6.9 and Corollary 6.10 imply Lippu f j Θpaq q ď C}f j } 2 C 1 Mpaq for every class a P PsefpXq; hence

Lip ´uf j ΘpG j`1,n T s σ n ω q ¯ď C}f j } 2 C 1 MpG j`1,n T s σ n ω q ď C}f j } 2 C 1 › › G j`1,n › › (6.34) ď C}f j } 2 C 1 n´1 ź "j`1 }f ˚ } H 1,1 ď C n´1 ź "j }f } 2 C 1 . (6.35)
Finally, since 1 ď Lippf j q for every 0 ď j ď n ´1, we obtain

Lip ´uf j ΘpG j`1,n T s σ n ω q ˝f j ω ¯ď Lip ´uf j ΘpG j`1,n T s σ n ω q ¯j´1 ź "0 Lippf q ď C n´1 ź "0 }f } 2 C 1 . (6.36)
Denoting the modulus of continuity by modcpu, rq " sup dpx,x 1 qďr |upxq ´upx 1 q|, we infer from Equation (6.33) that (6.37)

modcpu T s ω , rq ď 1 M ppf n ω q ˚pT s σ n ω qq ˜Cn n´1 ź "0 }f } 2 C 1 ¨r `› › u T s σ n ω › › 8 ¸.
To ease notation set λ " λ H 1,1 . Fix a small ε ą 0. By Lemma 5.13, for almost every ω there exists C " C ε pωq such that M ppf n ω q ˚pT s σ n ω qq ´1 ď Ce ´npλ´εq for every n. Fix M larger than but close to exp pE plog }f } C 1 qq. Applied to the ν N -integrable function ω " pf n q Þ Ñ log }f 0 } C 1 , the Birkhoff ergodic theorem gives (6.38)

n´1 ź "0 }f } 2 C 1 ď CM n as well as n n´1 ź "0 }f } 2 C 1 ď CM n
for some C " C M pωq (increase M to deduce the second inequality from the first). Thus, modcpu T s ω , rq ď C 3 e ´npλ´εq pM n r `eεn q " C 3 e ´npλ´2εq `pM e ´εq n r `1˘.

(
Choosing n so that r -pM e ´εq ´n we get modcpu T s ω , rq ď C 4 r θ with θ " λ´2ε log M `ε and the proof of the theorem is complete.

GLOSSARY OF RANDOM DYNAMICS, II

In this section we consider a random holomorphic dynamical system pX, νq on a compact Kähler surface, satisfying the moment condition (4.1). Our goal is to collect a number of facts from the ergodic theory of random dynamical systems, including the construction of associated skew products, fibered entropy and Lyapunov exponents of stationary measures, stable and unstable manifolds, and various measurable partitions. Here the group Γ ν may a priori be elementary; also, the compactness assumption on X can be dropped in most of these results (in this case (4.1) should be strengthened to a C 2 -moment condition). Since some subsequent arguments rely on the work [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF] of Brown and Rodriguez-Hertz, we have tried to make notation consistent with that paper as much as possible.

7.1. Skew products and stationary measures associated to pX, νq. Define:

-Ω " AutpXq N , whose elements are denoted by ω " pf n q ně0 . On Ω, the one-sided shift is denoted by σ : Ω Ñ Ω. -Σ " AutpXq Z , whose elements are denoted by ξ " pf n q nPZ . On Σ, the two-sided shift is denoted by ϑ : Σ Ñ Σ. -X " Σ ˆX and X `" Ω ˆX, whose elements are denoted by x " pξ, xq and

x " pω, xq respectively. The natural projections are denoted by π Σ : X Ñ Σ (resp.

π Ω : X `Ñ Ω) and π X : X Ñ X (resp. π X : X `Ñ X, using the same notation).

Recall that the product σ-algebra on Ω (resp. Σ) is generated by cylinders ( 3 ), and that it coincides with the Borel σ-algebra BpΩq (resp. BpΣq) (see [14, Lem. 6.4.2]).

7.1.1. Skew products. For ω P Ω and n ě 1, f n ω is the left composition f n ω " f n´1 ˝¨¨¨˝f 0 ; in particular, f 1 ω " f 0 (see § 4.3). For n " 0, we set f 0 ω " id. This is consistent with the notation used in the previous sections. The same notation f n ξ is used for ξ P Σ and n ě 0. When n ă 0, we set f n ξ " pf n q ´1 ˝¨¨¨˝pf ´1q ´1. With this definition the cocycle formula

f n`m ξ " f n ϑ m ξ ˝f m ξ
holds for all pm, nq P Z 2 and ξ P Σ. By definition, the skew products induced by the random dynamical system pX, νq are the transformations F `: X `Ñ X `and F : X Ñ X defined by (7.1)

F `: pω, xq Þ ÝÑ pσω, f 1 ω pxqq and F : pξ, xq Þ ÝÑ pϑξ, f 1 ξ pxqq. If : X Ñ X `
denotes the natural projection, then ˝F " F `˝ . Note that F is invertible, with F ´1px q " pϑ ´1ξ, f ´1 ϑ ´1ξ pxqq, but F `is not; indeed pX , F q is the natural extension of pX `, F `q.

Lemma 7.1. The measure µ on X is stationary if and only if the product measure m `:" ν N ˆµ on X `is invariant under F `.

A stationary measure is said ergodic if it is an extremal point in the convex set of stationary measures; hence, µ is ergodic if and only if m `is F `-ergodic. Actually µ is ergodic if and only if every ν-almost surely invariant measurable subset A Ă X (that is a measurable subset such that for ν-almost every f , µpA∆f ´1pAqq " 0) has measure µpAq " 0 or 1. This is by no means obvious since F `-invariant sets have no reason to be of product type. This statement is part of the so-called random ergodic theorem (see Propositions 1.8 and 1.9 in [START_REF] Benoist | Random walks on reductive groups, volume 62 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]). Proposition 7.2. There exists a unique F -invariant probability measure m on X projecting on m `under the natural projection X Ñ X `. Moreover, (1) the measure m is equal to the weak-‹ limit m " lim nÑ8 pF n q › `νZ ˆµ˘.

(2) the projections pπ Σ q ˚m and pπ X q ˚m are respectively equal to ν Z and µ;

(3) the equality m " ν Z ˆµ holds if and only if µ is f -invariant for ν-almost every f ; (4) pX , F, mq is ergodic if and only if pX `, F `, m `q is.

3 Cylinders are products C " ś Cj of Borel sets, all of which are equal to AutpXq except finitely many of them. For simplicity, we denote a cylinder by C "

ś N j"0 Cj if C k " AutpXq for |k| ą N .
The existence and uniqueness of m, as well as the characterization of its ergodicity, follow from the fact that pX , F q is the natural extension of pX `, F `q (see [81, §1.2] for a detailed explanation). See [91, §I.1] for the proof of Assertions (1), ( 2) and (3). 7.1.2. Past, future, and partitions. Let F denote the σ-algebra on X obtained by taking the mcompletion of BpΣq b BpXq. It will often be important to detect objects depending only on the "future" or on the "past". To formalize this, we define two σ-algebras on Σ:

-F`i s the ν Z -completion of the σ-algebra generated by the cylinders C " ś N j"0 C j . -F´i s the ν Z -completion of the σ-algebra generated by the cylinders C " ś ´1 j"´N C j .

To formulate it differently, we define local stable and unstable sets for the shift ϑ:

(7.2) Σ s loc pξq " tη P Σ ; @i ě 0, η i " ξ i u and Σ u loc pξq " tη P Σ ; @i ă 0, η i " ξ i u . Then a subset of Σ is F`measurable (resp. F´m easurable) if, up to a set of zero ν Z -measure, it is Borel and saturated by local stable sets Σ s loc pξq (resp. unstable sets Σ u loc pξq). The σ-algebra F `on X will be the m-completion of F`b BpXq. An F `-measurable object should be understood as "depending only on the future", thus it makes sense on X and on X `. Actually F coincides with the completion of the pull-back of BpX `q under : X Ñ X `. The σ-algebra F ´of "objects depending only on the past" is defined analogously. Consider the partition into the subsets F ´px q :" Σ u loc pξq ˆtxu (each of them can be naturally identified to Ω). Then, modulo m-negligible sets, the elements of F ´are saturated by this partition.

For ξ P Σ we set X ξ " tξu ˆX " π ´1 Σ pξq, which can be naturally identified with X via π X . The disintegration of the probability measure m with respect to the partition into fibers of π Σ gives rise to a family of conditional probabilities m ξ such that m " ş m ξ dν Z pξq, because pπ Σ q ˚m " ν Z . pf n ϑ ´nξ q › µ.

In particular, the family of measures ξ Þ Ñ m ξ is F ´-measurable.

Indeed, the convergence is a consequence of the martingale convergence theorem (see [12, §2.5] for details) and the second assertion easily follows.

Since ξ Þ Ñ m ξ is F ´-measurable, the conditional measures of m on the atoms F ´px q " Σ u loc pξq ˆtxu of the partition generating F ´are induced by the lifts of the conditionals of ν Z on the Σ u loc pξq, via the natural projection π Σ : X Ñ Σ. In addition we can simultaneously identify Σ u loc pξq to Ω and ν Z p ¨| Σ u loc q to ν N . In this way we get (7.3) mp ¨| F ´px qq " ν Z p ¨| Σ u loc pξqq ˆδx » ν N for m-almost every x " pξ, xq P X . This corresponds to Equation ( 9) in [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF]. By [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF]Prop. 4.6], this implies that F `X F ´is equivalent, modulo m-negligible sets, to tH, Σu b BpXq.

Lyapunov exponents.

Let µ be a stationary measure for pX, νq; assume that µ (or equivalently m or m `) is ergodic. The upper and lower Lyapunov exponents λ `ě λ ´are respectively defined by the almost sure limits

(7.4) λ `" lim nÑ8 1 n log }D x f n ω } and λ ´" lim nÑ8 1 n log › › ›pD x f n ω q ´1› › › ´1;
the existence of these limits is guaranteed by Kingman's subadditive ergodic theorem, thanks to the moment condition (4.1), and the convergence also holds on average. Let us now apply the Oseledets theorem successively to the tangent cocycle defined by the fiber dynamics pX `, F `, m `q, and then to the cocycle associated to pX , F, mq.

7.2.1. The non-invertible setting. Define the tangent bundles T X `:" Ω ˆT X and T X :" Σ ˆT X, and denote by DF and DF `the natural tangent maps, that is D pξ,xq F : tξu ˆTx X Ñ tϑξu ˆTf ξ pxq X is induced by D x f 1 ξ :

(7.5) D pξ,xq F pvq " D x f 1 ξ pvq p@v P T x X ξ " T x Xq For the non-invertible dynamics on X `, the Oseledets theorem gives: for m `-almost every pω, xq, there exists a non-trivial complex subspace V ´pω, xq of tωu ˆTx X such that

@v P V ´pω, xqzt0u, lim nÑ`8 1 n log }D x f n ω pvq} " λ (7.6) @v R V ´pω, xq, lim nÑ`8 1 n log }D x f n ω pvq} " λ `. (7.7)
The field of subspaces V ´is measurable and almost surely invariant. Two cases can occur: either λ ´ă λ `and V ´pω, xq is almost surely a complex line, or λ ´" λ `and V ´pω, xq " tωu ˆTx X.

7.2.2.

The invertible setting. For the dynamical system F : X Ñ X , the statement is:

-if λ ´" λ `then for m-almost every x " pξ, xq, for every non-zero v P T x X ξ » T x X,

(7.8) lim nÑ˘8 1 n log › › D x f n ξ pvq › › " λ
´;

-if λ ´ă λ `then for m-almost every x there exists a decomposition T x X ξ " E ´pξ, xq' E `pξ, xq such that for ‹ P t´, `u and every v P E ‹ pξ, xqz t0u, (7.9)

lim nÑ˘8 1 n log › › D x f n ξ pvq › › " λ ‹ .
Furthermore the line fields E ˘are measurable and invariant, and log |=pE ´, E `q| is integrable (here, the "angle" =pE ´px q, E `px qq is the distance between the two lines E ´px q and E `px q in PpT x X q). [START_REF] Crauel | Non-Markovian invariant measures are hyperbolic[END_REF], and also Avila-Viana [1, Thm B]).

Remark 7.5. If λ ´and λ `are both positive then µ is atomic. Indeed, since µ is almost surely invariant we get m " ν Z ˆµ. Reversing time, the Lyapunov exponents of m become negative, so as explained above the measures m ξ are atomic. By invariance m ξ " µ, so µ is atomic too.

By definition, µ is hyperbolic if λ ´ă 0 ă λ `. In this case we rather use the conventional superscripts s{u instead of ´{`for stable and unstable objects. We also have E s " V s in this case (and more generally when λ ´ă λ `); so, it follows that the complex line field E s on T X is F `-measurable. Conversely the unstable line field E u is F ´-measurable.

Invariant volume forms.

Let us start with a well-known result.

Lemma 7.6. Let pX, νq be a random holomorphic dynamical system satisfying the integrability condition (4.1), and µ be an ergodic stationary probability measure. Then

λ ´`λ `" ż log |Jac f pxq| dµpxqdνpf q,
where Jac denotes the Jacobian determinant relative to any smooth volume form on X.

We omit the proof, since this result is a corollary of Proposition 7.8 below. When X is an Abelian, or K3, or Enriques surface, Remark 3.15 provides an AutpXq-invariant volume form on X. Thus, we obtain: Corollary 7.7. Assume that X is an Abelian, or K3, or Enriques surface. Let ν be a probability measure on AutpXq satisfying the integrability condition (4.1), and µ be an ergodic ν-stationary measure. Then λ ´`λ `" 0.

Let η be a non-trivial meromorphic 2-form on the surface X. There is a cocycle Jac η , with values in the multiplicative group MpXq ˆof non-zero meromorphic functions, such that (7.10) f ˚η " Jac η pf qη for every f P AutpXq. We say that η is almost invariant if |Jac η pf qpxq| " 1 for every x P X and ν-almost every f P AutpXq (in particular Jac η pf q is a constant). We refer to §3. [START_REF] Baragar | Automorphisms of surfaces in a class of Wehler K3 surfaces with Picard number 4[END_REF] for examples with an invariant meromorphic 2-form.

Proposition 7.8. Let pX, νq be a random holomorphic dynamical system satisfying the integrability condition (4.1), and µ be an ergodic stationary measure. Let η be a non-trivial meromorphic 2-form on X such that

(i) ż log `| Jac η pf qpxq|dµpxqdνpf q ă `8;
(ii) µ gives zero mass to the set of zeroes and poles of η.

Then

λ ´`λ `" ż logp|Jac η f pxq| 2 qdµpxqdνpf q;

in particular λ ´`λ `" 0 if η is almost invariant.

Proof. Fix a trivialization of the tangent bundle T X, given by a measurable family of linear isomorphisms Lpxq : T x X Ñ C 2 such that (a) detpLpxqq " 1 and (b) 1{C ď }Lpxq} › › Lpxq ´1› › ď C, for some constant C ą 1; here, the determinant is relative to the volume form vol on X and the standard volume form dz 1 ^dz 2 on C 2 , and the norm is with respect to the Kähler metric pκ 0 q x on T x X and the standard euclidean metric on C 2 . For pξ, xq P X and n ě 0, the differential D x f n ξ is expressed in this trivialization as a matrix A pnq pξ, xq " Lpf n ξ pxqq ˝Dx f n ξ ˝Lpxq ´1. Let χ ń pξ, xq ď χ ǹ pξ, xq be the singular values of A pnq pξ, xq. Then m-almost surely, 1 n log χ n pξ, xq Ñ λ ˘as n Ñ `8. The form η ^η can be written η ^η " ϕpxqvol for some function ϕ : X Ñ r0, `8s. Locally, one can write η " hpxqdx 1 ^dx 2 where px 1 , x 2 q are local holomorphic coordinates and h is a meromorphic function; then ϕpxqvol " |hpxq| 2 dx 1 ^dx 2 ^dx 1 ^dx 2 . The jacobian Jac η satisfies (7.11) | Jac η pf qpxq| 2 " ϕpf pxqq ϕpxq Jac vol pf qpxq for every f P AutpXq and x P X. Using detpLpxqq " 1, we get (7.12) detpA pnq pξ, xqq " Jac vol pf n ξ qpxq, and then (7.13)

1 n log χ ń pξ, xq `1 n log χ ǹ pξ, xq " 2 n log ˇˇJac η f n ξ pxq ˇˇ´1 n logpϕpf n ξ pxqq{ϕpxqq.
By the Oseledets theorem, the left hand side of (7.13) converges almost surely to λ ´`λ `. Since the Jacobian Jac η is multiplicative along orbits, i.e. Jac η f n ξ pxq " Let divpηq be the set of zeroes and poles of η. Since µ is ergodic and does not charge divpηq, we deduce that for m-almost every pξ, xq, there is a sequence pn j q such that f n j ξ pxq stays at positive distance from divpηq; along such a sequence, log |ϕpf n j ξ pxqq{ϕpxq| stays bounded, and the right hand side of (7.13) tends to 2 ş log |Jac η f pxq| dµpxqdνpf q. This concludes the proof.

ś n´1 k"0 Jac η f ϑ k ξ pf k ξ xq
7.4. Intermezzo: local complex geometry. Recall that X is endowed with a Riemannian structure, hence a distance, induced by the Kähler metric κ 0 . For x P X, we denote by euc x the translation-invariant Hermitian metric on T x X (which is considered here as a manifold in its own right) associated to the Riemannian structure induced by pκ 0 q x . Given any orthonormal basis pe 1 , e 2 q of T x X for this metric, we obtain a linear isometric isomorphism from T x X to C 2 , endowed respectively with euc x and the standard euclidean metric; we shall implicitly use such identifications in what follows. We denote by Dpz; rq the disk of radius r around z in C, and set Dprq " Dp0; rq.

7.4.1.

Hausdorff and C 1 -convergence. Let U Ă C be a domain. If γ : U Ñ X is a holomorphic curve, we can lift it canonically to a curve γ p1q : U Ñ T X by setting γ p1q pzq " pγpzq, γ 1 pzqq P T γpzq X, where γ 1 pzq denotes the velocity of γ at z. The Kähler form κ 0 induces a Riemannian metric and therefore a distance dist T X on T X. We say that two parametrized curves γ 1 and γ 2 are δ-close in the C 1 -topology if dist T X pγ p1q 1 pzq, γ p1q 2 pzqq ď δ uniformly on U . This implies that γ 1 pU q and γ 2 pU q are δ-close in the Hausdorff sense, but the converse does not hold (take U " Dp1q, γ 1 pzq " pz, 0q, and γ 2 pzq " pz k , εz q with k and large while ε is small). 7.4.2. Good charts. Let R 0 be the injectivity radius of κ 0 . We fix once and for all a family of charts Φ x : U x Ă T x X Ñ X with the following properties (for some constant C 0 ): (i) Φ x p0q " x and pD Φ x q 0 " id TxX ; (ii) Φ x is a holomorphic diffeomorphism from its domain of definition U x to an open subset V x contained in the ball of radius R 0 around x; (iii) on U x , the Riemannian metrics euc x and Φ x κ 0 satisfy C ´1 0 ď euc x { Φ x κ 0 ď C 0 ; (iv) the family of maps Φ x depends continuously on x.

With r 0 ď R 0 {p ? 2C 0 q, we can add:

(v) for every orthonormal basis pe 1 , e 2 q of T x X, the bidisk Dpr 0 qe 1 `Dpr 0 qe 2 is contained in U x ; in particular, the ball of radius r 0 centered at the origin for euc x is contained in U x .

To make assertion (iv) more precise, fix a continuous family of orthonormal basis pe 1 pxq, e 2 pxqq on some open set V of X: Assertion (iv) means that, if we compose Φ x with the linear isomorphism pz 1 , z 2 q P C 2 Þ Ñ z 1 e 1 pxq `z2 e 2 pxq P T x X we obtain a continuous family of maps. If needed, we can also add the following property (see [71, pp. 107-109]):

(iii') euc x osculates Φ x κ 0 up to order 2 at x.

Families of disks.

A holomorphic disk ∆ Ă X containing x is said to be a disk of size (at least) r at x (resp. of size exactly r at x), for some r ă r 0 , if there is an orthonormal basis pe 1 , e 2 q of T x X such that Φ ´1 x p∆q contains (resp. is) the graph tze 1 `ϕpzqe 2 ; z P Dprqu for some holomorphic map ϕ : Dprq Ñ Dprq. By the Koebe distortion theorem if ∆ has size r at x, then its geometric characteristics around x at scale smaller than r{2, say, are comparable to that of a flat disk. An alternative definition for the concept of disks of size ě r could be that ∆ contains the image of an injective holomorphic map γ : Dprq Ñ X such that γpBDprqq Ă XzB X px; rq and }γ 1 } ď D, for some fixed constant D. Then, if ∆ contains a disk of size r for one of these definitions, it contains a disk of size ε 0 r for the other definition, for some uniform ε 0 ą 0; in particular, there is a constant C depending only on pX, κ 0 q such that a disk of size r at x contains an embedded submanifold of B X px; Crq.

Let px n q be a sequence converging to x in X, and let r be smaller than the radius r 0 introduced in Assertion (v), § 7.4.2. Let ∆ n be a family of disks of size at least r at x n and ∆ be a disk of size at least r at x. We say that ∆ n converges towards ∆ as a sequence of disks of size r, if there is an orthonormal basis pe 1 , e 2 q of T x X for euc x such that (i) Φ ´1 x p∆q contains the graph tze 1 `ϕpzqe 2 ; z P Dprqu for some holomorphic function ϕ : Dprq Ñ Dprq; (ii) for every s ă r, if n is large enough, the disk Φ ´1 x p∆ n q contains the graph tze 1 φn pzqe 2 ; z P Dpsqu of a holomorphic function ϕ n : Dpsq Ñ Dprq; (iii) for every ε ą 0, we have |ϕpzq ´ϕn pzq| ă ε on Dpsq if n is large enough.

By the Cauchy estimates, the convergence then holds in the C 1 -topology (see § 7.4.1). It follows from the usual compactness criteria for holomorphic functions that the space of disks of size r on X is compact (for the topology induced by the Hausdorff topology in X). Likewise, if a sequence of disks of size r converges in the Hausdorff sense, then it also converges in the C 1 sense, at least as disks of size s ă r, because two holomorphic functions ϕ and ψ from Dprq to Dprq whose graphs are ε-close are also εpr ´sq ´1-close in the C 1 -topology.

It may also be the case that the ∆ n are contained in different fibers X ξn of X . By definition, we say that the sequence ∆ n converges to ∆ Ă X ξ if ξ n converges to ξ and the projections of ∆ n converge to ∆ in X.

7.4.4. Entire curves. An entire curve in X is, by definition, a holomorphic map ψ : C Ñ X. The curve is immersed if its velocity ψ 1 does not vanish. Our main examples of immersed curves will, in fact, be injective and immersed entire curves. If ψ 1 and ψ 2 are two immersed entire curves with the same image, there exists a holomorphic diffeomorphism of C, i.e. a nonconstant affine map A : z Þ Ñ az `b, such that ψ 2 " ψ 1 ˝A. If ψ is an immersed entire curve and |ψ 1 | ě η on Dpz 0 , sq, its image contains a disk of size Cs at ψpz 0 q, for some C ą 0 that depends only on η and κ 0 . 7.5. Stable and unstable manifolds. By Lemma 4.1, Condition (4.1) implies similar moment conditions for higher derivatives, so Pesin's theory applies. The following proposition summarizes the main properties of Pesin local stable and unstable manifolds. Recall that a function h is ε-slowly varying, relatively to some dynamical system g, if e ´ε ď hpgpxqq{hpxq ď e ε for every x. We view the stable manifold of x " pξ, xq as contained in X ξ ; it can also be viewed as a subset of X: whether we consider one or the other point of view should be clear from the context. If x " pξ, xq and y " pξ, yq are points of the same fiber X ξ , we denote by dist X px , y q the Riemannian distance between x and y computed in X. Proposition 7.9. Let pX, νq be a random holomorphic dynamical system, and µ be an ergodic and hyperbolic stationary measure. Then, for every δ ą 0, there exists measurable positive δ-slowly varying functions r and C on X (depending on δ) and, for m-almost every x " pξ, xq P X , local stable and unstable manifolds W s rpx q px q and W u rpx q px q in X ξ such that m-almost surely:

(1) W s rpx q px q and W u rpx q px q are holomorphic disks of size at least 2rpx q at x respectively tangent to E s px q and E u px q;

(2) for every y P W s rpx q px q and every n ě 0, dist X pF n px q, F n py qq ď Cpx q expppλ ´`δqnq;

likewise for every y P W u rpx q px q and every n ě 0 dist X pF ´npx q, F ´npy qq ď Cpx q expp´pλ `´δqnq;

(3) F pW s rpx q px qq Ă W s rpF px qq pF px qq and F ´1pW u rpF px qq pF px qqq Ă W u rpx q px q.

By Lusin's theorem, for every ε ą 0 we can select a compact subset R ε Ă X with mpR ε q ą 1 ´ε, on which rpx q and Cpx q can be replaced by uniform constants (respectively denoted by r and C) and the following additional property holds:

(4) on R ε the local stable and unstable manifolds W s{u r px q vary continuously for the C 1topology (in the sense of § 7.4.1 and 7.4.3).

The subsets R ε are usually called Pesin sets, or regular sets. We also denote the local stable or unstable manifolds by W s{u loc px q, or by W s{u r px q when x is in a Pesin set on which rp¨q ě r. On several occasions we will have to deal with measurability issues for W s{u loc px q as a function of x : this will be done by exhausting X by Pesin sets and using their continuity on R ε .

The global stable and unstable manifolds of x are respectively defined by the following increasing unions:

(7.15) W s px q " ď ně0 F ´n ´W s rpx q pF n px qq ¯and W u px q " ď ně0 F n ´W u rpx q pF ´npx qq ¯.
In particular, they are injectively immersed holomorphic curves in X ξ . Pesin theory shows that:

W s px q " " pξ, yq P X ξ ; lim sup nÑ8 1 n log dist X pF n pξ, yq, F n pξ, xqq ă 0 * (7.16) W u px q " " pξ, yq P X ξ ; lim sup nÑ´8 1 |n| log dist X pF n pξ, yq, F n pξ, xqq ă 0 * . ( 7
.17) Proposition 7.10. Under the assumptions of Proposition 7.9, W s px q and W u px q are biholomorphic to C for m-almost every x .

More precisely, W s px q is parametrized by an injectively immersed entire curve ψ s

x : C Ñ X such that ψ s

x p0q " x and this parametrization is unique, up to an homothety z Þ Ñ az of C. Likewise, W u px q is parametrized by such an entire curve ψ u

x .

Proof. By (7.15) and Proposition 7.9.(3), W s px q is an increasing union of disks and is therefore a Riemann surface homeomorphic to R 2 ; so, it is biholomorphic to C or D. Let A Ă X be a set of positive measure on which r ě r 0 and C ď C 0 . By Proposition 7.9.(2), there exists n 0 P N and m 0 ą 0 such that if n ě n 0 and if x and F n px q belong to A, then W s r pF n pξ, xqqz pF n W s r pξ, xqq is an annulus of modulus ě m 0 . Now for m-almost every x P X there is an infinite sequence pk j q such that F k j px q P A and k j`1 ´kj ą n 0 . For such an x , W s px qzW s r px q contains an infinite nested sequence of annuli of modulus at least m 0 , namely the F ´kj`1 pW s r pF k j`1 px qqzF k j`1 ´kj pW s r pF k j px qq. Thus, W s px q is biholomorphic to C.

If we are only interested in stable manifolds, there is a simplified version of Proposition 7.9 which takes place on X: Proposition 7.11. Let pX, νq be a random holomorphic dynamical system and µ an ergodic stationary measure, whose Lyapunov exponents satisfy λ ´ă 0 ď λ `. Then for m `-almost every pω, xq the stable set

W s pω, xq " " y P X ; lim sup nÑ8 1 n log dist X pf n ω pyq, f n ω pxqq ă 0 * is an injectively immersed entire curve in X.
Indeed, stable manifolds can be obtained from a purely "one-sided" construction, that is, by considering only positive iterates (see [START_REF] Liu | Smooth ergodic theory of random dynamical systems[END_REF]Chap. III]). This also shows that local stable manifolds in X are F `-measurable, and may be viewed as living in X `.

Fibered entropy.

Here we recall the definition of the metric fibered entropy of a stationary measure µ (see [81, §2.1] or [91, Chap. 0 and I] for more details). If η is a finite measurable partition of X, its entropy relative to µ is H µ pηq " ´řCPη µpCq log µpCq. Then, we set

h µ pX, ν; ηq " lim nÑ8 1 n ż H µ ˜n´1 ł k"0 ´f k ξ ¯´1
pηq ¸dν N pξq, (7.18) h µ pX, νq " sup th µ pX, ν; ηq ; η a finite measurable partition of Xu . (7.19) Actually h µ pX, ν; ηq can be interpreted as a conditional (or fibered) entropy for the skewproducts F `on X `and F on X . Indeed, the so-called Abramov-Rokhlin formula holds [START_REF] Bogenschütz | The Abramov-Rokhlin formula[END_REF]:

h µ pX, νq " h ν N ˆµpF `|η Ω q " h m `pF `q ´hν N pσq (7.

20)

" h m pF |η Σ q " h m pF q ´hν Z pϑq, (7.21) where η Ω (resp. η Σ ) denotes the partition into fibers of the first projection π Ω : X `Ñ Ω (resp. π Σ : X Ñ Σ) and in the second and fourth equalities we assume h ν N pσq " h ν Z pϑq ă 8. The next result is the fibered version of the Margulis-Ruelle inequality. Proposition 7.12. Let pX, νq be a random holomorphic dynamical system satisfying the moment condition (4.1) and µ be an ergodic stationary measure. If h µ pX, νq ą 0 then µ is hyperbolic and minpλ `, ´λ´q ě 1 2 h µ pX, νq.

Proof. See [START_REF] Bahnmüller | A Margulis-Ruelle inequality for random dynamical systems[END_REF] or [91, Chap. II] for the inequality λ `ě 1 2 h µ pX, νq. For ´λ´ě 1 2 h µ pX, νq, we use the fact that h m pF |η Σ q " h m pF ´1|η Σ q (see e.g. [91, I.4.2]) and apply the Margulis-Ruelle inequality to F ´1. Beware that there is a slightly delicate point here: pF ´1, mq is not associated to a random dynamical system in our sense; fortunately, the statement of the Margulis-Ruelle inequality in [START_REF] Bahnmüller | A Margulis-Ruelle inequality for random dynamical systems[END_REF] (see also [91, Appendix A]) covers this situation. 7.7. Unstable conditionals and entropy. Assume µ is ergodic and hyperbolic. By definition, an unstable Pesin partition η u on X is a measurable partition of pX , F, µq with the following properties:

η is increasing: F ´1η u refines η u ; -for m-almost every x , η u px q is an open subset of W u px q and (7.22)

ď ně0 F n `ηu pF ´npx qq ˘" W u px q; -η u is a generator, i.e. Ž 8
n"0 F ´npη u q coincides m-almost surely with the partition into points.

Here, as usual, η u px q denotes the atom of η u containing x , and F ´1η u is the partition defined by pF ´1η u qpx q " F ´1pη u pF px qqq. The definition of a stable Pesin partition η s is similar. A neat proof of the existence of such a partition is given by Ledrappier and Strelcyn in [START_REF] Franc | A proof of the estimation from below in Pesin's entropy formula[END_REF], which easily adapts to the random setting (see [START_REF] Liu | Smooth ergodic theory of random dynamical systems[END_REF]§IV.2], and [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]). Lemma 7.13. There exists a stable (resp. unstable) Pesin partition whose atoms are F `measurable (resp. F ´-measurable), that is, saturated by local stable (resp. unstable) sets Σ s loc ˆtxu (resp. Σ u loc ˆtxu).

The existence of unstable partitions enables us to give a meaning to the unstable conditionals of m. Indeed, first observe that if η u and ζ u are two unstable Pesin partitions, then m-almost surely mp¨|η u q and mp¨|ζ u q coincide up to a multiplicative factor on η u px q X ζ u px q. Furthermore, there exists a sequence of unstable partitions η u n such that for almost every x , if K is a compact subset of W u px q for the intrinsic topology (i.e. the topology induced by the biholomorphism W u px q » C) then K Ă η u n px q for sufficiently large n: indeed by (7.22), the sequence of partitions F n η u does the job. Hence almost surely the conditional measure of m on W u px q is well-defined up to scale; we define m u

x by normalizing so that m u x pη u px qq " 1. The next proposition is known as the (relative) Rokhlin entropy formula, stated here in our specific context. Proposition 7.14. Let pX, νq be a random holomorphic dynamical system satisfying the moment condition (4.1), and µ be an ergodic and hyperbolic stationary measure. Let η u be an unstable Pesin partition. Then

h µ pX, νq " H m pF ´1η u | η u q :" ż log J η u px qdmpx q,
where J η u px q is the "Jacobian" of F relative to η u , that is

J η u px q " m `F ´1 pη u pF px qqq | η u px q ˘´1 .
Sketch of proof. The argument is based on the following sequence of equalities, in which η Σ is the partition into fibers of π Σ , as before:

h µ pX, νq " h m pF |η Σ q " h m pF ´1|η Σ q " h m pF ´1|η u _ η Σ q (7.23) :" H m pη u |F η u _ η Σ q " H m pη u |F η u q " H m pF ´1η u |η u q
The equalities in the first and last line follow from the general properties of conditional entropy: see [91, Chap. 0] for a presentation adapted to our context (note that the conditional entropy would be denoted by h η Σ m there) or Rokhlin [START_REF] Rokhlin | Lectures on the entropy theory of transformations with invariant measure[END_REF] for a thorough treatment. On the other hand the equality (7.23) is non-trivial. If η u were of the form Ž `8 n"0 η, where η is a 2-sided generator with finite entropy, this equality would follow from the general theory. For a Pesin unstable partition the result was established for diffeomorphisms in [START_REF] Franc | The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin's entropy formula[END_REF]Cor 5.3] and adapted to random dynamical systems in [91, Cor. VI.7.1].

Remark 7.15. It is customary to present the Rokhlin entropy formula using unstable partitions, mostly because entropy is associated to expansion. Nonetheless, a similar formula holds in the stable direction:

h µ pX, νq " ż log J η s px qdmpx q where J η s px q " m `F `ηs pF ´1px qq ˘| η s px q ˘´1 .
The proof is identical to that of Proposition 7.14, applied to F ´1, with however the same caveat as in Proposition 7.12: pF ´1, mq is not associated to a random dynamical system in our sense.

The only non-trivial point is to check that the key equality (7.23) holds in this case. Fortunately, the main purpose of [START_REF] Bahnmüller | Characterization of measures satisfying the Pesin entropy formula for random dynamical systems[END_REF] is to explain how to adapt [91, Chap. VI], hence the equality (7.23), to a more general notion of "random dynamical system" which covers the case of pF ´1, mq (see in particular the last lines of [3, §5] for a short discussion of the Rokhlin formula).

The following consequence of the Rokhlin formula will play an important role in Section 9.

Corollary 7.16. Under the assumptions of the previous proposition, the following assertions are equivalent:

(a) h µ pX, νq " 0;

(b) mp¨|η u px qq " δ x for m-almost every x ; (c) mp¨|η u px qq is atomic for m-almost every x .

The same result holds for the stable Pesin partition η s .

Proof. In view of the definition of J η u , the entropy vanishes if and only if for m-almost every x , mp¨|η u px qq is carried by a single atom of the finer partition F ´1η u . Now since H m pF ´1η u | η u q "

1 n H m pF ´nη u | η u q, the same is true for F ´nη u , and finally since pF ´nη u q is generating, we conclude that (a)ô(b). That (c) implies (a) follows from the same ideas but it is slightly more delicate, see [111, §2.1-2.2] for a clear exposition in the case of the iteration a single diffeomorphism, which readily adapts to our setting. The result for the stable Pesin partition η s follows by changing F to F ´1 (see however Remark 7.15).

STABLE MANIFOLDS AND LIMIT CURRENTS

Let as before pX, νq be a non-elementary random holomorphic dynamical system on a compact Kähler (hence projective) surface, and assume µ is an ergodic stationary measure admitting exactly one negative Lyapunov exponent, as in Proposition 7.11. Our purpose in this section is to relate the stable manifolds W s pω, xq to the stable currents T s ω constructed in §6. According to Proposition 7.11, the stable manifolds are parametrized by injective entire curves; the link between these curves and the stable currents will be given by the well-known Ahlfors-Nevanlinna construction of positive closed currents associated to entire curves. 8.1. Ahlfors-Nevanlinna currents. We denote by tV u the integration current on a (possibly non-closed, or singular) curve V . Let φ : C Ñ X be an entire curve. By definition, if α is a test 2-form, xφ ˚tDp0, tqu , αy " ş Dp0,tq φ ˚α, which accounts for possible multiplicities coming from the lack of injectivity of φ; φ ˚tDp0, tqu " tφpDp0, tqqu when φ is injective. Set for R ą 0. When φ is an immersion, ApRq is the area of φpDp0, Rqq; in all cases, ApRq is the mass of φ ˚tpDp0, Rqqu.

Proposition 8.1 (see Brunella [22, §1]). If φ : C Ñ X is a non-constant entire curve, there exist sequences of radii pR n q increasing to infinity such that the sequence of currents

N pR n q " 1 T pR n q ż Rn 0 φ ˚tDp0, tqu dt t
converges to a closed positive current T . If furthermore φpCq is Zariski dense, and T is such a closed current, the class rT s P H 1,1 pX, Rq is nef. In particular xrT s | rT sy ě 0 and xrT s | rCsy ě 0 for every algebraic curve C Ă X.

Such limit currents T will be referred to as Ahlfors-Nevanlinna currents associated to the entire curve φ : C Ñ X. If φpCq is not Zariski dense then the closure φpCq (for the euclidean topology) is a (possibly singular) curve of genus 0 or 1; if φ is injective, then φpCq is rational. 8.2. Equidistribution of stable manifolds. If µ is hyperbolic, or more generally if it admits exactly one negative Lyapunov exponent, then, for m `-almost every x " pω, xq P X `, the stable manifold W s px q, which is viewed here as a subset of X as in Proposition 7.11, is parametrized by an injectively immersed entire curve. Then we can relate the Ahlfors-Nevanlinna currents to the limit currents T s ω ; here are the three main results that will be proved in this section. Theorem 8.2. Let pX, νq be a non-elementary random holomorphic dynamical system on a compact Kähler surface, satisfying (4.1). Let µ be an ergodic stationary measure such that λ ´pµq ă 0 ď λ `pµq. Then exactly one of the following alternative holds.

(a) For m `-almost every x , the stable manifold W s px q is not Zariski dense. Then µ is supported on a Γ ν -invariant curve Y Ă X and for m `-almost every x , W s px q Ă Y . In addition every component of Y is a rational curve, and the intersection form is negative definite on the subspace of H 1,1 pX; Rq generated by the classes of components of Y . (b) For m `-almost every x the stable manifold W s px q is Zariski dense and the only normalized Ahlfors-Nevanlinna current associated to W s px q is T s ω . Corollary 8.3. Under the assumptions of Theorem 8.2, if in addition µ is hyperbolic and nonatomic, then the Alternative (b) is equivalent to

(b') µ is not supported on a Γ ν -invariant curve.
Corollary 8.4. Under the assumptions of Theorem 8.2, assume furthermore that ν satisfies the exponential moment condition (5.23). Then in Alternative (b) there exists θ ą 0 such that for m `-almost every x P X `the Hausdorff dimension of W s px q equals 2 `θ.

8.3.

Proof of Theorem 8.2 and its corollaries. We work under the assumptions of Theorem 8.2.

Lemma 8.5. If there exists a proper Zariski closed subset of X with positive µ-measure, then:

either µ is the uniform counting measure on a finite orbit of Γ ν ; -or µ has no atom and it is supported on a Γ ν -invariant algebraic curve, which is the Γ ν -orbit of an irreducible algebraic curve.

Proof. Consider the real number δ 0 max pµq " max xPX µ ptxuq. If δ 0 max pµq ą 0, there is a nonempty finite set F Ă X for which µ ptxuq " δ 0 max pµq. By stationarity, F is Γ ν -invariant, and by ergodicity µ is the uniform measure on F . Now, assume that µ has no atom. Let δ 1 max pµq be the maximum of µpDq among all irreducible curves D Ă X. If µpZq ą 0 for some proper Zariski closed subset Z Ă X, then δ 1 max pµq ą 0. Since two distinct irreducible curves intersect in at most finitely many points and µ has no atom, there are only finitely many irreducible curves E such that µpEq " δ 1 max pµq. To conclude, we argue as in the zero dimensional case.

If V Ă X is a smooth curve, possibly with boundary, if T is a closed positive p1, 1q-current on X with a continuous normalized potential u T (as in § 6.1.1), then by definition (8.2) xT ^tV u , ϕy "

ż V ϕ ΘpT q `żV ϕ dd c pu T | V q,
for every test function ϕ. Here is the key relation between stable manifolds and limit currents:

Lemma 8.6. For m `-almost every x " pω, xq, if ∆ is a disk contained in W s px q, then T s ω t∆u " 0.

Proof. With no loss of generality we assume that the boundary of the disk ∆ in W s px q » C is smooth. We consider points x " pω, xq P X `which are generic in the following sense: they are regular from the point of view of Pesin's theory, and T s ω satisfies the conclusions of §6. By Pesin's theory, for every ε ą 0, there is a set A ε Ă N of density larger than 1 ´ε, such that for n in A ε , the local stable manifold W s r pF n `px qq is a disk of size r " rpεq at f n ω pxq and f n ω p∆q is a disk contained in an exponentially small neighborhood of f n ω pxq. We have

(8.3) MpT s σ n ω ^tf n ω p∆quq " ż W s r pF n `px qq 1 f n ω p∆q ΘpT s σ n ω q `żW s r pF n `px qq 1 f n ω p∆q dd c u T s σ n ω .
Since MpT s σ n ω q " 1, Lemma 6.1 shows that ΘpT s σ n ω q is bounded by Aκ 0 ; so the first integral on the right hand side of (8.3) is bounded by a constant times the area of f n ω p∆q, which is exponentially small. By ergodicity, there exists

A 1 ε Ă A ε of density at least 1 ´2ε such that if n P A 1 ε , }u T s σ n ω } 8 is bounded by some contant D ε ą 0.
For such an n, let χ be a test function in W s r pF n `px qq such that χ " 1 in W s r{2 pF n `px qq, and vanishing near BW s r pF n `px qq. Note that since W s r pF n `px qq is of size r, the C 2 -norm of χ depends only on r. We write ż

W s r pF n `px qq 1 f n ω p∆q dd c u T s σ n ω ď ż W s r pF n `px qq χdd c u T s σ n ω " ż W s r pF n `px qq u T s σ n ω dd c χ (8.4) ď Cprq}χ} C 2 › › u T s σ n ω › › 8
where Cprq bounds the area of W s r pF n `px qq; this last term is uniformly bounded because n P A 1 ε . Thus we conclude that MpT s σ n ω ^tf n ω p∆quq is bounded along such a subsequence. On the other hand, the relation pf n ω q ˚T s σ n ω " Mppf n ω q ˚T s σ n ω qT s ω gives (8.5) T s σ n pωq ^tf n ω p∆qu " M ´pf n ω q ˚T s σ n pωq ¯pf n ω q ˚pT s ω ^t∆uq.

The mass Mppf n ω q ˚pT s ω ^t∆uqq is constant, equal to the mass of the measure T s ω ^t∆u; so

(8.6) M ´T s σ n pωq ^tf n ω p∆qu ¯" Mppf n ω q ˚T s σ n pωq qMpT s ω ^t∆uq.
By Lemma 5.13, Mppf n ω q ˚T s σ n pωq q goes exponentially fast to infinity. Since the left hand side is bounded, this shows that MpT s ω ^t∆uq " 0, as desired.

With Lemma 2.13, the following statement takes care of the first alternative in Theorem 8.2.

Lemma 8.7. If there is a Borel subset A Ă X `of positive measure such that for every x P A, the stable manifold W s px q is contained in an algebraic curve, then µ is supported on a Γ νinvariant algebraic curve. In addition, for m `-almost every x , W s px q is an irreducible rational curve of negative self-intersection.

Proof. For x P A, let Dpx q be the Zariski closure of W s px q. Discarding a set of measure zero if needed, W s px q is biholomorphic to C so Dpx q is a (possibly singular) irreducible rational curve, and Dpx qzW s px q is reduced to a point. By Lemma 8.6, T s ω ^t∆u " 0 for every disk ∆ Ă W s px q. Since T s ω has continuous potentials, T s ω ^tDpx qu gives no mass to points (see e.g. [START_REF] Cantat | Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy[END_REF]Lem. 10.13] for the singular case). It follows that T s ω ^tDpx qu " 0, hence xepωq | rDpx qsy " 0.

By the Hodge index theorem, either rDpx qs 2 ă 0 or rDpx qs is proportional to epωq, however this latter case would contradict the fact that epωq is ν N -almost surely irrational (see Theorem 5.7; one could also use that Curpepωqq is reduced to T s ω ). Thus, rDpx qs 2 ă 0. An irreducible curve with negative self-intersection is uniquely determined by its cohomology class; since NSpX; Zq is countable, there are only countably many irreducible curves pD k q kPN with negative self intersection. Since W s loc px q Ă D k if and only if Dpx q " D k , and since local stable manifolds vary continuously on the Pesin regular set R ε for every ε ą 0, we infer that tx P A ; Dpx q " D k u is measurable for every k. Hence there exists an index k such that m `ptx P A ; rDpx qs " rD k suq ą 0. Since x belongs to W s loc px q, Fubini's theorem implies that µpD k q ą 0, and Lemma 8.5 shows that µ is supported on the Γ ν -orbit of D k .

Finally, this argument shows that the property W s loc px q Ă Ť kPN D k , or equivalently that W s loc px q is contained in a rational curve of negative self intersection, is invariant and measurable, so by ergodicity of m `it is of full measure. The proof is complete.

We are now ready to conclude the proof of Theorem 8.2. Let A be the set of Pesin regular points such that W s px q is contained in an algebraic curve. From the proof of Lemma 8.7, x belongs to A if and only if W s loc px q is contained in one of the countably many irreducible curves D k Ă X of negative self-intersection. This condition determines a countable union of closed subsets in the Pesin sets R ε , hence A is Borel measurable. By Lemma 8.7, if A has positive m `-measure then Alternative (a) holds. So, if (a) is not satisfied, W s px q is almost surely Zariski dense. Pick such a generic x , which further satisfies the conclusion of Lemma 8.6, and let N be an Ahlfors-Nevanlinna current associated to W s px q. By Proposition 8.1, rN s is a nef class so rN s 2 ě 0. Thus, if we are able to show that xrN s | rT s ω sy " 0, we deduce from the Hodge index theorem and MpN q " 1 that rN s " rT s ω s " epωq, hence N " T s ω by Theorem 6.11. So, it only remains to prove that xrN s | rT s ω sy " 0, or equivalently (8.7)

N ^T s ω " 0. This is intuitively clear because N is an Ahlfors-Nevanlinna current associated to the entire curve W s px q and T s ω ^t∆u " 0 for every bounded disk ∆ Ă W s px q. However, there is a technical difficulty to derive (8.7) from T s ω ^t∆u " 0, even if W s px q is an increasing union of such disks ∆.

At least two methods were designed to deal with this situation: the first one uses the geometric intersection theory of laminar currents (see [7,[START_REF] Dujardin | Sur l'intersection des courants laminaires[END_REF]), and the second one was developed by Dinh and Sibony in the preprint version of [START_REF] Dinh | Green currents for holomorphic automorphisms of compact Kähler manifolds[END_REF] (details are published in [33, §10.4]). Unfortunately these papers only deal with the case of currents of the form lim n 1 ApRnq φpDp0, R n qq, instead of the Ahlfors-Nevanlinna currents introduced in Section 8.1, which were designed to get the nef property stated in Proposition 8.1. So, we have to explain how to adapt the formalism of [7,[START_REF] Dujardin | Sur l'intersection des courants laminaires[END_REF] to the Ahlfors-Nevanlinna currents of Proposition 8.1.

Following [START_REF] Duval | Singularités des courants d'Ahlfors[END_REF] we say that T is an Ahlfors current if there exists a sequence p∆ n q of unions of smoothly bounded holomorphic disks such that lengthpB∆ n q " o pMp∆ n qq and T is the limit as n Ñ 8 of the sequence of normalized integration currents 1 Mp∆nq t∆ n u; here, lengthpB∆ n q is by definition the sum of the lengths of the boundaries of the disks constituting ∆ n , lengths which are computed with respect to the Riemannian metric induced by κ 0 . We say furthermore that T is an injective Ahlfors current if the disks constituting ∆ n are disjoint or intersect along subsets with relative non-empty interior. By discretizing the integral defining the currents N pR n q in Proposition (8.1) we see that any Ahlfors-Nevanlinna current is an injective Ahlfors current.

Strongly approximable laminar currents are a class of positive currents introduced in [56] with geometric properties which are well suited for geometric intersection theory. In a nutshell, a current T is a strongly approximable laminar current if for every r ą 0, there exists a uniformly laminar current T r (non closed in general) made of disks of size r, and such that MpT ´Tr q " Opr 2 q. This mass estimate is crucial for the geometric understanding of wedge products of such currents. Since these notions have been studied in a number of papers, we refer to [7,[START_REF] Dujardin | Sur l'intersection des courants laminaires[END_REF][START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF] for definitions, the basic properties of these currents, and technical details. This presentation in terms of disks of size r is from [57, §4]. The next lemma is a mild generalization of the methods of [7, §7], [24, §4.3] and [56, §4]. For completeness we provide the details in Appendix B. Lemma 8.8. Any injective Ahlfors current T on a projective surface X is a strongly approximable laminar current: if T " lim n 1 Mp∆nq t∆ n u where the disks ∆ n have smooth boundaries and lengthpB∆ n q " o pMp∆ n qq, one can construct a family of uniformly laminar currents T r , whose constitutive disks are limits of pieces of the ∆ n , and such that if S is any closed positive current with continuous potential on X, then S ^Tr increases to S ^T as r decreases to 0.

With this lemma at hand, let us conclude the proof of Theorem 8.2. Since X is projective, we can apply the previous lemma to any Ahlfors-Nevanlinna current N associated to W s px q. In this way we get a family of currents N r such that N r ^T s ω increases to N ^T s ω as r decreases to 0. On the other hand, by Lemma 8.6, the intersection of T s ω with every disk contained in W s px q vanishes, so again using the fact that T s ω has a continuous potential, we infer that if ∆ is any disk subordinate to N r , T s ω ^t∆u " 0. Hence N r ^T s ω " 0 for every r ą 0, and finally N ^T s ω " 0, as desired.

Proof of Corollary 8.3. Since (b') and (a) are contradictory, (b') implies (b). Conversely assume that µ is hyperbolic, non atomic and supported on a Γ ν -invariant curve C. Since µ has no atom, it gives full mass to the regular set of C, hence Σ ˆT pRegpCqq defines a DF -invariant bundle, and by the Oseledets theorem the ergodic random dynamical system pC, ν, µq must either have a positive or a negative Lyapunov exponent. If this exponent were positive then µ would be atomic, as observed in Section 7.2.3. Hence, the Lyapunov exponent tangent to C is negative and W s px q is contained in C for m `-almost every x . So (b) implies (b').

Proof of Corollary 8.4. Since ν satisfies an exponential moment condition, Theorem 6.16 provides a θ ą 0 such that u T s ω is Hölder continuous of exponent θ for ν N -almost every ω. This implies that T s ω gives mass 0 to sets of Hausdorff dimension ă 2 `θ (see [START_REF] Sibony | Dynamique des applications rationnelles de P k . In Dynamique et géométrie complexes[END_REF]Thm 1.7.3]). Since for m `-almost every x, SupppT s ω q Ă W s px q, we infer that HDim `W s px q ˘ě 2 `θ. To conclude the proof it is enough to show that x Þ Ñ HDim `W s px q ˘is constant on a set of full m `-measure. Indeed, x Þ Ñ HDim `W s px q ˘defines an F `-invariant function, defined on the full measure set R of Pesin regular points. If we show that this function is measurable, then the result follows by ergodicity. This is a consequence of the following two facts:

(1) the assignment x Þ Ñ W s px q defines a Borel map from R to the space KpXq of compact subsets of X;

(2) the function KpXq Q K Þ Ñ HDimpKq is Borel (see [95, Thm 2.1]).
In both cases KpXq is endowed with the topology induced by the Hausdorff metric. For the first point, observe that R is the increasing union of the compact sets R ε so it is Borel; then, on a Pesin set R ε , x Þ Ñ W s r px q is continuous, so x Þ Ñ F ´n`W s r pF n px qq ˘is continuous as well. Since F ´n`W s r pF n px qq ˘converges to W s px q in the Hausdorff topology, we infer that x Þ Ñ W s px q is a pointwise limit of continuous maps on R ε , hence Borel, and finally x Þ Ñ W s px q is Borel on R, as claimed.

NO INVARIANT LINE FIELDS

As above, let pX, νq be a random holomorphic dynamical system satisfying the moment condition (4.1), and µ be an ergodic hyperbolic stationary measure. From §7.2 and §7.5, the local stable manifolds and stable Oseledets directions are F `-measurable; so, E s pξ, xq is naturally identified to E s pω, xq under the projection pξ, xq P X Þ Ñ pω, xq P X `, and the same property holds for stable manifolds. Then, m `-almost every x P X `has a Pesin stable manifold W s px q (resp. direction E s px q). Let V px q " V pω, xq be such a measurable family of objects (stable manifolds, or stable directions, etc); we say that V px q is non-random if for µ-almost every x, V pω, xq does not depend on ω, that is, there exists V pxq such that V pω, xq " V pxq for ν Nalmost every ω. If V is not non-random, we say that V depends non-trivially on the itinerary. Since stable directions depend only on the future, the random versus non-random dichotomy can be analyzed in X `or in X . Our purpose in this section is to establish the following result. Theorem 9.1. Let pX, νq be a non-elementary random holomorphic dynamical system on a compact Kähler surface satisfying the Condition (4.1). Let µ be an ergodic and hyperbolic stationary measure, not supported on a Γ ν -invariant curve. Then the following alternative holds:

(a) either the Oseledets stable directions depend non-trivially on the itinerary; (b) or µ is ν-almost surely invariant and h µ pX, νq " 0.

In fact, our stiffness theorems imply that often, µ is also invariant in case (a) (see §10). In (b), the almost-sure invariance implies that µ is in fact Γ ν -invariant (see Remark 4.2). It turns out that (a) and (b) are mutually exclusive. Indeed the main argument of [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF] (4 ) implies that the fiber entropy is positive if the Oseledets stable directions depend non-trivially on the itinerary (see [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF]Rmk 12.3]). So we get the following: Corollary 9.2. Let pX, ν, µq be as in Theorem 9.1. If µ is not ν-almost surely invariant, then its fiber entropy is positive. 9.1. Intersection multiplicities. If V 1 and V 2 are germs of curves at 0 P C 2 , with an isolated intersection at 0, the intersection multiplicity inter 0 pV 1 , V 2 q is, by definition, the number of intersection points of V 1 and V 2 `u in N for small generic u P C 2 , where N is a neighborhood of 0 such that V 1 XV 2 XN " t0u (see [39, §12]). It is a positive integer, and inter 0 pV 1 , V 2 q " 1 if and only if V 1 and V 2 are transverse at 0. We extend this definition by setting inter 0 pV 1 , V 2 q " 0 if V 1 or V 2 does not contain 0 and inter 0 pV 1 , V 2 q " 8 if 0 is not an isolated point of V 1 X V 2 , that is locally V 1 and V 2 share an irreducible component. The intersection multiplicity extends to analytic cycles (that is, formal integer combinations of analytic curves).

Lemma 9.3. The multiplicity of intersection inter 0 p¨, ¨q is upper semi-continuous for the Hausdorff topology on analytic cycles.

In our situation we will only apply this result to holomorphic disks with multiplicity 1, in which case the topology is just the usual local Hausdorff topology.

Proof. Assume inter 0 pV 1 , V 2 q " k and V 1,n Ñ V 1 (resp. V 2,n Ñ V 2 )
as cycles; we have to show that lim sup inter 0 pV 1,n , V 2,n q ď k. If k " 8 there is nothing to prove. Otherwise, t0u is isolated in V 1 X V 2 , so we can fix a neighborhood U of 0 such that V 1 X V 2 X U " t0u; then, the result follows from [39, Prop 2 p.141] (stability of proper intersections). 9.2. Generic intersection multiplicity of stable manifolds. Recall from §7.5 that for m-almost every x " pξ, xq P X there exists a local stable manifold W s rpx q px q Ă X ξ » X, depending measurably on x ; we might simply denote it by W s loc px q. Let us cover a subset of full measure in X by Pesin subsets R εn . Take a point x P X, and consider the set of points ppξ, xq, pζ, xqq P R εn ˆRεm , for some fixed pair of indices pn, mq; Lemma 9.3 shows that the intersection multiplicity inter x pW s loc pξ, xq, W s loc pζ, xqq is an upper semi-continuous function of ppξ, xq, pζ, xqq on that compact set. Thus, the intersection multiplicity inter x pW s loc pξ, xq, W s loc pζ, xqq is a measurable function of pξ, ζq. Recall that -the σ-algebra F ´on X is generated, modulo m-negligible sets, by the partition into subsets of the form Σ u loc pξq ˆtxu (see § 7.1, Equation (7.2)); Proof. The relation defined on X by pξ, xq » k pη, yq if x " y and W s loc pξ, xq and W s loc pη, yq have order of contact at least k `1 at x is an equivalence relation which defines a partition Q k of X . We shall see below that Q k is a measurable partition. Since F : X Ñ X acts by diffeomorphisms on the fibers X of X , we get that F pQ k px qq " Q k pF px qq for almost every x P X . Then, the proof of [21, Lemma 9.9] applies verbatim to show that if

-ξ Þ Ñ m ξ is F ´-measurable, i
(9.2) m ` x ; mpQ k px q|F ´px qq ą 0 (˘ą 0, then (9.3) m ` x ; mpQ k px q|F ´px qq " 1 (˘" 1.
This is exactly the desired statement. (This assertion says more than the mere ergodicity of m, which only implies that m ptx ; mpQ k px q|F ´px qq ą 0uq " 1.) It remains to explain why Q k is a measurable partition. For this, we have to express the atoms of Q k as the fibers of a measurable map to a Lebesgue space. As for the measurability of the intersection multiplicity, we consider an exhaustion of X by countably many Pesin sets; then, it is sufficient to work in restriction to some compact set K Ă X on which local stable manifolds have uniform size and vary continuously. Taking a finite cover of X by good charts (see § 7.4.2), and restricting K again to keep only those local stable manifolds which are graphs over some fixed direction, we can also assume that π X pKq is contained in the image of a chart Φ x 0 : U x 0 Ñ V x 0 Ă X and there is an orthonormal basis pe 1 , e 2 q such that for every y P K the local stable manifold π X pW s loc py qq is a graph tze 1 `ψs y pzqe 2 u in this chart, for some holomorphic function ψ s y on Dprq. Now the map from K to C 2 ˆCk defined by (9.4) x Þ ÝÑ ´Φ´1

x 0 pπ X px qq, pψ s x q 1 p0q, . . . , pψ s x q pkq p0q īs continuous. Since the fibers of this map are precisely the (intersection with K of the) atoms of Q k , we are done.

The previous lemma is stated on X because its proof relies on the ergodic properties of F . However, since stable manifolds depend only on the future, it admits the following more elementary formulation on X: Corollary 9.5. Let k ě 1 be an integer. Exactly one of the following assertions holds:

(a) for µ-almost every x P X and pν N q 2 -almost every pω, ω 1 q, inter x `W s loc pω, xq, W s loc pω 1 , xq ˘ě k `1; (b) or for µ-almost every x P X and pν N q 2 -almost every pω, ω 1 q, inter x `W s loc pω, xq, W s loc pω 1 , xq ˘ď k.

Combined with results from the previous sections, this alternative leads to the existence of a finite order of contact k 0 between generic stable manifolds W s loc pω, xq and W s loc pω 1 , xq: Lemma 9.6. There exists a unique finite integer 1 ď k 0 ă `8 such that for µ-almost every x P X and pν N q 2 -almost every pair pω, ω 1 q, inter x pW s pω, xq, W s pω 1 , xqq " k 0 .

Proof. Fix a small ε ą 0 and consider a compact set R ε Ă X `with m `pR ε q ě 1 ´ε, along which local stable manifolds have size at least rpεq and vary continuously. Since by Theorem 8.2 for m `-a.e. x , the only Nevanlinna current associated to W s px q is T s ω , we can further assume that this property holds for every x P R ε . Let A Ă X be a subset of full µ-measure on which the alternative of Corollary 9.5 holds for every k ě 1. In X `, consider the measurable partition into fibers of the form Ω ˆtxu; it corresponds to the partition F ´in Lemma 9.4. Then, the associated conditional measures m `p ¨| Ω ˆtxuq are naturally identified with ν N . Fix x P A such that m `pR ε |Ωˆtxuq ą 0. Since pX, νq is non-elementary, Theorems 5.7 and 6.11 provide pairs pω 1 , ω 2 q in pπ Ω pR ε qq 2 for which the currents T s ω 1 and T s ω 2 are not cohomologous. By Theorem 8.2 these currents describe respectively the asymptotic distribution of W s pω 1 , xq and W s pω 2 , xq so we infer that W s pω 1 , xq ‰ W s pω 2 , xq and by the analytic continuation principle it follows that W s loc pω 1 , xq ‰ W s loc pω 2 , xq. Let k 1 ă 8 be the intersection multiplicity of these manifolds at x. Since the intersection multiplicity is upper semi-continuous, we infer that for ω 1 j P R ε close to ω j , j " 1, 2, inter x pW s loc pω 1 1 , xq, W s loc pω 1 2 , xqq ď k 1 . Thus for k " k 1 we are in case (b) of the alternative of Corollary 9.5. Applying then Corollary 9.5 successively for k " 1, . . . , k 1 , there is a first integer k 0 for which case (b) holds, and since (a) holds for k 0 ´1, we conclude that generically inter x pW s loc pω, xq, W s loc pω 1 , xqq " k 0 . 9.3. Transversal perturbations. The key ingredient in the proof of Theorem 9.1 is the following basic geometric lemma, which is a quantitative refinement of [7,Lemma 6.4].

Lemma 9.7. Let k be a positive integer. If r and c are positive real numbers, then there are two positive real numbers δ " δpk, r, cq and α " αpk, r, cq with the following property. Let M 1 and M 2 be two complex analytic curves in Dprq ˆDprq Ă C 2 such that (i) M 1 and M 2 are graphs tpz, f j pzqq ; w P D r u of holomorphic functions f j : Dprq Ñ Dprq;

(ii) M 1 X M 2 " tp0, 0qu, and inter p0,0q pM 1 , M 2 q " k;

(iii) the k-th derivative satisfies ˇˇpf 1 ´f2 q pkq p0q ˇˇě c.

If M 3 Ă Dprq ˆDprq is a complex curve that does not intersect M 1 but is δ-close to M 1 in the C 1 -topology , then M 2 and M 3 have exactly k transverse intersection points in Dpαrq ˆDpαrq (i.e. with multiplicity 1).

Proof. Without loss of generality we may assume that δ ă 1.

Step 1.-We claim that there exists α 1 " α 1 pk, r, cq such that for every α ď α 1 and every z P Dpαrq the following estimates hold: There exists α 1 pk, r, cq such that as soon as α ď α 1 , the right hand side of this inequality is smaller than c |z| k {p2k!q; hence Estimate (9.5) follows. The same argument applies for (9.6) because ˇˇˇˇg 1 pzq ´gpkq p0q pk ´1q! z k´1 ˇˇˇˇď 4pk `1q

1 2 ˇˇpf 1 ´f2 q pkq p0q ˇǩ! |z| k ď |f 1 pzq ´f2 pzq| ď 3 2 ˇˇpf 1 ´f2 q pkq p0q ˇǩ! |z| k (9.
ˆ|z| r ˙k ˆ1 ´|z| r ˙´2 ď 4pk `1qr 1´k α p1 ´αq 2 |z| k´1 .
Step 2.-For every α ď α 1 , if δ ă cpαrq k {2k!, M 2 and M 3 have exactly k intersection points, counted with multiplicities, in Dpαrq ˆDpαrq.

Indeed, the intersection points of M 3 and M 2 correspond to the solutions of the equation f 3 " f 2 . To locate its roots, note that on the circle BDpαrq, the Inequality (9.5) implies (9.7)

|f 1 ´f2 | ě 1 2 c k! pαrq k .
Since |f 1 ´f3 | ă δ, the choice δ ă cpαrq k {2k! is tailored to assure that the hypothesis of the Rouché theorem is satisfied in Dpαrq; so, counted with multiplicities, there are k solutions to the equation f 3 " f 2 on that disk. Furthermore by the Schwarz lemma |f 2 | ă αr on Dpαrq so the corresponding intersection points between M 2 and M 3 are contained in Dpαrq ˆDpαrq.

If k " 1 the proof is already complete at this stage, so from now on we assume k ě 2.

Step 3.-Set δ 0 " |f 3 p0q|, and note that δ 0 ď δ. Then for every α ď 1{2, in Dpαrq we have in Dp2αrq, and (9.9) follows from the Cauchy estimate }g 1 } Dpαrq ď pαrq ´1}g} Dp2αrq .

Step 4.-We now conclude the proof. Fix α " αpk, r, cq such that α ď α 1 and (9.12) βpαq :"

1 ´2α 1 `2α ´k ´1 k ˆ1 `α 1 ´α ą 0.
(This will be our final choice for α.) Fix δ ă cpαrq k {2k! and consider a solution z 0 of the equation f 2 pzq " f 3 pzq in Dpαrq provided by Step 2. The transversality of M 2 and M 3 at pz 0 , f 2 pz 0 qq is equivalent to f 1 3 pz 0 q ‰ f 1 2 pz 0 q, so we only need (9.13) ˇˇpf 3 ´f1 q 1 pz 0 q ˇˇă ˇˇpf 2 ´f1 q 1 pz 0 q ˇˇ.

Since pf 1 ´f3 qpz 0 q " pf 1 ´f2 qpz 0 q, combining the right hand side of Inequality (9.5) and the left hand side of Inequality 9.8, we get that (9.14) 3 2

ˇˇpf 1 ´f2 q pkq p0q ˇǩ! |z 0 | k ě δ 1`α 1´α 0 . thus (9.15) |z 0 | ě δ 1 k 1`α 1´α 0 ˆ2k! 3 ˙1 k ˇˇpf 1 ´f2 q pkq p0q ˇˇ´1 k .
Hence by (9.6) we get that

ˇˇpf 2 ´f1 q 1 pz 0 q ˇˇě 1 2pk ´1q! ˆ2k! 3 ˙k´1 k δ k´1 k 1`α 1´α 0 ˇˇpf 1 ´f2 q pkq p0q ˇˇ1 k (9.16) ě 1 2pk ´1q! ˆ2k! 3 ˙k´1 k δ k´1 k 1`α 1´α 0 c 1 k .
On the other hand by Estimate (9.9) (9.17) ˇˇpf 3 ´f1 q 1 pz 0 q ˇˇď 1 αr δ 1´2α 1`2α 0 Since δ 0 ď δ, we only need to impose one more constraint on δ (together with δ ă cpαrq k {2k!), namely (9.18)

δ βpαq ă 1 2pk ´1q! ˆ2k! 3 ˙k´1 k c 1 k rα,
to get the desired inequality |pf 3 ´f1 q 1 pz 0 q| ă |pf 2 ´f1 q 1 pz 0 q|. . This function satisfies P 1 n p5{nq " 10{n, P n p5{nq " p5{nq 2 , and 15{n 2 ď P n ď 35{n 2 on r´1, 1s. Now, if n is large, M 1 " ty " 0u, M 2 " y " x 2 ( and M 3 " ty " P n pxqu are three smooth algebraic curves in p´1, 1q 2 Ă R 2 such that M 3 is disjoint from M 1 but close to it in the C 1 topology, and M 3 is tangent to M 2 at p5{n, 25{n 2 q. Similar arguments can be used to show that the semi-continuity of Lemma 9.3 fails for real analytic curves (though Corollary 9.5 may still be valid for real analytic random dynamical systems).

Let ∆ 1 and ∆ 2 be two disks of size r at x P X, which are tangent at x; let e 1 P T x X be a unit vector in T x ∆ 1 " T x ∆ 2 and e 2 a unit vector orthogonal to e 1 for κ 0 . Then, in the chart Φ x , ∆ 1 and ∆ 2 are graphs tze 1 `ψi pzqe 2 u of holomorphic functions ψ i : Dprq Ñ Dprq, i " 1, 2, such that ψ i p0q " 0 and ψ 1 i p0q " 0. If inter x p∆ 1 , ∆ 2 q " k, then for j " 1, . . . , k ´1 one has ψ pjq 1 p0q " ψ pjq 2 p0q and ψ pkq 1 p0q ‰ ψ pkq 2 p0q. We define the k-osculation of ∆ 1 and ∆ 2 at x to be (9.19) osc k,x,r p∆ 1 , ∆ 2 q " ˇˇψ pkq 1 p0q ´ψpkq 2 p0q ˇˇ.

If s ď r and we consider ∆ 1 and ∆ 2 as disks of size s, then osc k,x,s p∆ 1 , ∆ 2 q " osc k,x,r p∆ 1 , ∆ 2 q. Thus, osc k,x,r p∆ 1 , ∆ 2 q does not depend on r, so we may denote this osculation number by osc k,x p∆ 1 , ∆ 2 q. With this terminology, Lemma 9.7 directly implies the following corollary.

Corollary 9.9. Let k be a positive integer, and r and c be positive real numbers. Then, there are two positive real numbers δ and α, depending on pk, r, cq, satisfying the following property. Let ∆ 1 and ∆ 2 be two holomorphic disks of size r through x, such that inter x p∆ 1 , ∆ 2 q " k and osc k,x p∆ 1 , ∆ 2 qq ě c. Let ∆ 3 be a holomorphic disk of size r such that ∆ 3 is δ-close to ∆ 1 in the C 1 -topology but ∆ 3 X ∆ 1 " H. Then ∆ 3 intersects ∆ 2 transversely in exactly k points in B X px, αrq.

The following lemma follows directly from the first step of the proof of Lemma 9.7.

Lemma 9.10. Let k be a positive integer, and r and c be positive real numbers. Then there exists a constant β depending only on pr, k, cq such that if ∆ 1 and ∆ 2 are two holomorphic disks of size r through x, such that k " inter x p∆ 1 , ∆ 2 q and osc k,x p∆ 1 , ∆ 2 qq ě c, then x is the only point of intersection between ∆ 1 and ∆ 2 in the ball B X px, βrq. 9.4. Proof of Theorem 9.1. Before starting the proof, we record the following two facts from elementary measure theory:

Lemma 9.11. Let pΩ, F, Pq be a probability space, and δ P p0, 1q.

(1) If ϕ is a measurable function with values in r0, 1s and such that ş ϕ dP ě 1 ´δ, then

P ´!x ; ϕpxq ě 1 ´?δ
)¯ě 1 ´?δ.

(2) If A j is a sequence of measurable subsets such that PpA j q ě 1 ´δ for every j, then Pplim sup A j q ě 1 ´δ.

Let us now prove Theorem 9.1. If the integer k 0 of Lemma 9.6 is equal to 1, then Pesin stable manifolds corresponding to different itineraries at a µ-generic point x P X are generically transverse; hence, we are in case (a) of the theorem -note that the conclusion is actually stronger than mere non-randomness. So, we now assume k 0 ą 1 and we prove that µ is almost surely invariant and that its entropy is equal to zero.

Step 1.-First, we construct a subset G ε of "good points" in X . As described in Section 7.1.2, the atoms of F ´are the sets F ´px q " Σ u loc pξq ˆtxu and the measures mp ¨|F ´px qq can be naturally identified to ν N under the natural projections

F ´px q " Ñ Σ u loc pξq " Ñ Ω.
For notational simplicity we denote these measures by m F x . For a small ε ą 0, let R ε Ă X be a compact subset with mpR ε q ą 1 ´ε, along which local stable manifolds have size at least 2rpεq and vary continuously. Since ş m F x pR ε q dmpx q ě 1 ´ε, by Lemma 9.11 [START_REF] Avila | Extremal Lyapunov exponents: an invariance principle and applications[END_REF] we can select a compact subset R 1 ε Ă R ε with mpR 1 ε q ě 1 ´?ε such that for every x P R 1 ε one has m F x pR ε q ě 1 ´?ε. By assumption, inter x pW s loc py 1 q, W s loc py 2 qq " k 0 for m-almost every x " pξ, xq P R 1 ε and for pm F x b m F x q-almost every pair of points py 1 , y 2 q P pF ´px q X R ε q 2 . Then there exists R 2 ε Ă R 1 ε of measure at least 1 ´2? ε and a constant cpεq ą 0 such that (9.20) osc k 0 ,x,rpεq pW s loc py 1 q, W s loc py 2 qq ě cpεq for every x " pξ, xq P R 2 ε and all pairs py 1 , y 2 q in a subset A ε,x Ă pF ´px q X R ε q 2 depending measurably on x and of measure (9.21) pm F x b m F x qpA ε,x q ě 1 ´4? ε (we just used pm F x b m F x qppF ´px q X R ε q 2 q ě p1 ´?εq 2 ą 1 ´2? ε). Finally, Fubini's theorem and Lemma 9.11 (1) provide a set G ε Ă R 2 ε such that (a) mpG ε q ě 1 ´2ε 1{4 (b) for every x P G ε , W s loc px q has size 2rpεq; (c) for every x P G ε , there exists a measurable set G ε,x Ă F ´px q with m F x pG ε,x q ě 1 ´2ε 1{4 such that for every y in G ε,x , W s loc py q has size ě rpεq and, viewed as a subset of X, -it is tangent to W s loc px q to order k 0 at x, -osc k 0 ,x,rpεq pW s loc px q, W s loc py qq ě cpεq. Note that x R G ε,x : indeed, when the local stable manifolds vary continuously, one can think of A ε,x as the complement of a small neighborhood of the diagonal in Ω ˆΩ.

Step 2.-To make the argument more transparent, we first show that the fiber entropy vanishes.

W s px q W s pζ, xj q W s pξj , xj q x x xj W s px q FIGURE 1.
On the left, a generic point x with the local stable manifolds W s loc pξ i , xq for distinct pξ i q iě0 (see Step 1). On the right, the choice of the sequence pζ, x j q gives a family of local stable manifolds (see Step 2).

Let η s be a Pesin partition subordinate to local stable manifolds in X . By Corollary 7.16 it is enough to show that for m-almost every x , mp¨|η s px qq is atomic (hence concentrated at x). Assume by contradiction that this is not the case. Therefore for ε ą 0 small enough there exists x " pξ, xq P G ε such that mp¨|η s px qq |η s px qXGε is non-atomic, and there exists an infinite sequence of pairwise distinct points x j " pξ, x j q in G ε X η s px q converging to x . Then with G ε,‹ as in Property (c) of the definition of G ε , we have m F xj pG ε,x j q ě 1 ´2ε 1{4 for every j.

Identifying all F ´px j q with Σ u loc pξq, by Lemma 9.11 [START_REF] Bahnmüller | A Margulis-Ruelle inequality for random dynamical systems[END_REF] we can find ζ P Σ u loc pξq such that pζ, x j q belongs to G ε,pζ,x j q for infinitely many j's. Along this subsequence the local stable manifolds W s loc pζ, x j q form a sequence of disks of uniform size r " 2rpεq at x j . Two such local stable manifolds are either pairwise disjoint or coincide along an open subset because they are associated to the same itinerary ζ.

Let us now use the notation from Corollary 9.9 and Lemma 9.10. We know that W s rpεq pζ, x j q is tangent to W s rpεq pξ, xq at x j to order k 0 , with osc k 0 ,x j ,rpεq pW s rpεq px q, W s rpεq pζ, x j qq ě cpεq; so, by Lemma 9.10, W s rpεq pζ, x j q and W s rpεq pζ, x j 1 q are disjoint as soon as dist X px j , x j 1 q ă βrpεq. Finally, if j and j 1 are large enough, then dist X px j , x j 1 q ă αrpεq and the C 1 distance between W s rpεq pζ, x j q and W s rpεq pζ, x j 1 q is smaller than δ; thus, Corollary 9.9 asserts that W s rpεq pζ, x j q and W s rpεq pζ, x j 1 q cannot both be tangent to W s rpεq pξ, xq. This is a contradiction, and we conclude that the fiber entropy of m vanishes.

Step (this is identical to the argument of Corollary 7.16). In particular for small ε we can find x " pξ, xq P G ε and a sequence of points x j " pξ j , x j q P G ε such that x j belongs to Ppx q X G ε , x j ‰ x and px j q converges to x in X. We can also assume that the x j are all distinct. By definition of G ε , m F xj `Gε,x j ˘ě 1 ´2ε 1{4 for every j. For pξ, ζq P Σ 2 , set (9.25) rξ, ζs " Σ u loc pξq X Σ s loc pζq; that is, rξ, ζs is the itinerary with the same past as ξ and the same future as ζ. As above, identifying the atoms of the partition F ´with Ω, Lemma 9.11 (2) provides an infinite subsequence pj q and for every an itinerary ζ j P Σ u loc pξ j q such that y j :" pζ j , x j q belongs to G ε,x j and all the ζ j have the same future, that is ζ j is of the form rξ j , ζs for a fixed ζ. By definition, inter x j pW s loc px j q, W s loc py j qq " k 0 (9.26) osc k 0 ,x j ,rpεq pW s loc px j q, W s loc py j qq ě cpεq. (9.27)

In addition the disks π X pW s loc py j qq are pairwise disjoint or locally coincide because the x j are distinct and the ζ j have the same future. Moreover, since x j belongs to Ppx q, W s px j q coincides with W s px q. Therefore, the π X pW s loc py j qq form a sequence of disjoint disks of size 2rpεq at x j , all tangent to π X pW s loc px qq to order k 0 , with osculation bounded from below by cpεq. Since this sequence of disks is continuous and px j q converges towards x, Lemma 9.10 and Corollary 9.9 provide a contradiction, exactly as in Step 2. This completes the proof of the theorem.

STIFFNESS

Here we study Furstenberg's stiffness property for automorphisms of compact Kähler surfaces, thereby proving Theorem A. Our first results in §10.3 deal with elementary subgroups of AutpXq. The argument relies on the classification of such elementary groups together with general group-theoretic criteria for stiffness; these criteria are recalled in § 10.1 and 10.2. Theorem 10.10 concerns the much more interesting case of non-elementary subgroups; its proof combines all results of the previous sections with the work of Brown and Rodriguez-Hertz [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF]. 10.1. Stiffness. Following Furstenberg [START_REF] Furstenberg | Stiffness of group actions[END_REF], a random dynamical system pX, νq is stiff if any ν-stationary measure is almost surely invariant; equivalently, every ergodic stationary measure is almost surely invariant. This property can conveniently be expressed in terms of ν-harmonic functions on Γ. Indeed if ξ : X Ñ R is a continuous function and µ is ν-stationary, then Γ Q g Þ Ñ ş X ξpgxq dµpxq is a bounded, continuous, right ν-harmonic function on Γ; thus proving that µ is invariant amounts to proving that such harmonic functions are constant. Stiffness can also be defined for group actions: a group Γ acts stiffly on X if and only if pX, νq is stiff for every probability measure ν on Γ whose support generates Γ; in this definition, the measures ν can also be restricted to specific families, for instance symmetric finitely supported measures, or measures satisfying some moment condition. There are some general criteria ensuring stiffness directly from the properties of Γ. A first case is when G is a topological group acting continuously on X and Γ Ă G is relatively compact. Then Γ acts stiffly on X: this follows from the maximum principle for harmonic functions on Γ (see also [START_REF] Furstenberg | Stiffness of group actions[END_REF]Thm 3.5]). Another important case for us is that of Abelian and nilpotent groups.

Theorem 10.1. Let G be a locally compact, second countable, topological group. Let ν be a probability measure on G. If G is nilpotent of class ď 2, then any measurable, ν-harmonic, and bounded function ϕ : G Ñ R is constant; thus, every measurable action of such a group is stiff.

We only stated the simplest result sufficient for our paper, but this theorem holds for most nilpotent groups without any assumption on the nilpotent class. It is due to Dynkin and Malyutov for any finitely generated nilpotent group, and to Guivarc'h for a large class of locally compact nilpotent groups; the case of Abelian groups is the famous Blackwell-Choquet-Deny theorem. We refer to [START_REF] Guivarc | Croissance polynomiale et périodes des fonctions harmoniques[END_REF] for a proof (6 ). We shall apply Theorem 10.1 to subgroups A Ă AutpXq; what we implicitly do is first replace A by its closure in AutpXq to get a locally compact group, and then apply the theorem to this group. 10.2. Subgroups and hitting measures. A basic tool is the hitting measure on a subgroup, which we briefly introduce now (see [START_REF] Benoist | Random walks on reductive groups, volume 62 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]Chap. 5] for details). Let G be a locally compact second countable topological group. A notion of length can be defined in this context as follows: given a neighborhood U of the unit element, for any g P G, length U pgq is the least integer n ě 1 such that g P U n . By definition a probability measure ν on G has a finite first moment (resp. a finite exponential moment) if ş length U pgq dνpgq ă 8 (resp. if ş exppα length U pgqq dνpgq ă 8 for some α ą 0). This condition does not depend on the choice of U .

Let ν be a probability measure on G, and consider the left random walk on G governed by ν. Given a subgroup H Ă G, for ω " pg i q P G N , define the hitting time (10.1)

T pωq " T H pωq :" min tn ě 1 ; g n ¨¨¨g 1 P Hu .

If T is almost surely finite we say that H is recurrent and the distribution of g T pωq ¨¨¨g 1 is by definition the hitting measure of ν on H, which will be denoted ν H . The key property of ν H is that if ϕ : G Ñ R is a ν-harmonic function, then ϕ| H is also ν H -harmonic. Therefore, if µ is a ν-stationary measure, then it is also ν H -stationary. Conversely, any bounded ν H -harmonic function h on H admits a unique extension r h to a bounded ν-harmonic function on G; this extension is defined by the formula (10.2) r hpxq " E x phpg T x,H pωq ¨¨¨g 1 xqq "

ż hpg T x,H pωq ¨¨¨g 1 xq dν N pωq
where the stopping time T x,H is defined by T x,H pωq " min tn ě 0 ; g n ¨¨¨g 1 x P Hu. The uniqueness comes from Doob's optional stopping theorem, which asserts that if pM t q tě0 is a bounded martingale and T is a stopping time which is almost surely finite then EpM T q " EpM 0 q. Thus, any bounded ν-harmonic function h on G satisfies Formula (10.2). If rG : Hs ă 8 then H is recurrent and its stopping time admits an exponential moment. It follows that ν H has a finite first (resp. exponential) moment if and only if ν does.

Likewise, assume that H is a normal subgroup of G with G{H isomorphic to Z, and that ν is symmetric with a finite first moment. Then, the projection ν of ν on G{H is symmetric with a finite first moment, so the random walk governed by ν on G{H » Z is recurrent (see the Chung-Fuchs Theorem in [58, §5.4] or [START_REF] Chung | On the distribution of values of sums of random variables[END_REF]) and H is recurrent. Lemma 10.2. Let ν be a probability measure on AutpXq and Γ 1 be a closed subgroup which is recurrent for the random walk induced by ν. Let ν 1 be the induced measure on Γ 1 . If pX, ν 1 q is stiff then pX, νq is stiff as well. This holds in particular if:

(i) either rΓ ν : Γ 1 s ă 8 (ii) or Γ 1 is a normal subgroup of Γ ν with Γ ν {Γ 1 isomorphic to Z, and ν is symmetric with a finite first moment.

Proof. Let µ be a ν-stationary measure on X. Then µ is ν 1 -stationary, hence by stiffness it is Γ 1 -invariant. Therefore for every Borel set B Ă X, the function Γ Q g Þ Ñ µpg ´1Bq is a bounded ν-harmonic function which is constant on Γ 1 so by the uniqueness of harmonic extension it is constant, and ν is Γ-invariant. [START_REF] Favre | Le groupe de Cremona et ses sous-groupes de type fini[END_REF]): if Γ ν is elementary and Γ ν is infinite there exists a finite index subgroup A ˚Ă Γ ν which is (a) either cyclic and generated by a loxodromic map; (b) or a free Abelian group of parabolic transformations possessing a common isotropic line; in that case, there is a genus 1 fibration τ : X Ñ S, onto a compact Riemann surface S, such that Γ ν permutes the fibers of τ .

Denote by ρ Γν : Γ ν Ñ Γ ν the restriction of ρ to Γ ν . We distinguish two cases.

Proof when the kernel of ρ Γν is finite. Let A be the pre-image of A ˚in Γ; it fits into an exact sequence 1 Ñ F Ñ A Ñ A ˚Ñ 0 with F finite, so a classical group theoretic lemma (see Corollary 4.8 in [START_REF] Cantat | Elements generating a proper normal subgroup of the cremona group[END_REF]) asserts that A contains a finite index, free Abelian subgroup A 0 , such that ρ Γν pA 0 q has finite index in A ˚. Since A 0 is Abelian, Theorem 10.1 shows that the action of pA 0 , ν A 0 q on X is stiff. The index of A 0 in Γ being finite, Lemma 10.2 concludes the proof.

Proof when the kernel of ρ Γν is infinite. In case (a), X is a torus C 2 {Λ and kerpρ Γν q is a group of translations of X (see Proposition 3.16). Let A Ă Γ ν be the pre-image of A ˚; setting K " kerpρ Γν q, we obtain an exact sequence 0 Ñ K Ñ A Ñ A ˚Ñ 0, with A Ă Γ ν of finite index, A ˚» Z generated by a loxodromic element, and K Ă X an infinite group of translations. Since ν is symmetric, the measure ν A is also symmetric; since ν A satisfies the moment condition (4.1), its projection on A ˚has a first moment (note that if f is loxodromic, then logp}pf ˚qn }q -|n|).

Since K is Abelian, its action on X is stiff; thus, as in Lemma 10.2.(ii), the action of A on X is stiff. Since A has finite index in Γ, the action of Γ on X is stiff too by Lemma 10.2.(i).

In case (b), we apply Proposition 2.18. So, either X is a torus, or the action of Γ ν on the base S of its invariant fibration τ : X Ñ S has finite order. In the latter case, a finite index subgroup Γ 0 of Γ preserves each fiber of τ ; then, Γ 0 contains a subgroup of index dividing 12 acting by translations on these fibers. This shows that Γ is virtually Abelian; in particular, Γ is stiff. The last case is when the image of Γ in AutpSq is infinite and X is a torus C 2 {Λ X . Then, S " C{Λ S is an elliptic curve and τ is induced by a linear projection C 2 Ñ C, say the projection px, yq Þ Ñ x. Lifting Γ to C 2 , and replacing Γ by a finite index subgroup if necesssary, its action is by affine transformations of the form (10.4) f : px, yq Þ Ñ px `a, y `mx `bq with m in C ˚, and pa, bq in C 2 . This implies that Γ is a nilpotent group of length ď 2; by Theorem 10.1 it also acts stiffly and we are done.

Example 10.4. If X " P 2 pCq, its group of automorphism is PGL 3 pCq and for most choices of ν there is a unique stationary measure, which is not invariant; the dynamics is proximal, and this is opposite to stiffness (see [START_REF] Furstenberg | Stiffness of group actions[END_REF]). If X " P 1 pCq ˆC, for some algebraic curve C, then AutpXq contains PGL 2 pCq ˆAutpCq; if ν is a probability measure on PGL 2 pCq ˆtid C u, then in most cases the stationary measures are again non invariant.

Proposition 10.5. Let X be a complex projective surface, and Γ be a subgroup of AutpXq such that Γ ˚is finite. If Γ preserves a probability measure, whose support is Zariski dense in X, then the action of Γ on X is stiff.

The main examples we have in mind is when the invariant measure is given by a volume form, or by an area form on the real part XpRq for some real structure on X, with XpRq ‰ H.

Proof. Replacing Γ by a finite index subgroup we may assume that Γ Ă AutpXq ˝. Denote by µ the invariant measure. Let G be the closure (for the euclidean topology) of Γ in the Lie group AutpXq ˝; then G is a real Lie group preserving µ.

Let α X : X Ñ A X be the Albanese morphism of X. There is a homomorphism of complex Lie groups τ : AutpXq ˝Ñ AutpA X q ˝such that α X ˝f " τ pf q ˝αX for every f in AutpXq ˝.

Pick a very ample line bundle L on X, denote by P N pCq the projective space PpH 0 pX, Lq _ q, where N `1 " h 0 pX, Lq, and by Ψ L : X Ñ P N pCq the Kodaira-Iitaka embedding of X given by L. By hypothesis, pΨ L q ˚µ is not supported by a hyperplane of P N pCq.

Step 1.-Suppose τ pGq " 1. Since Pic 0 pXq and A X are dual to each other, G acts trivially on Pic 0 pXq and L is G-invariant, that is g ˚L " L for every g P G. Thus there is a homomorphism β : G Ñ PGL N `1pCq such that Ψ L ˝g " βpgq ˝ΨL for every g P L. If G is not compact, there is a sequence of elements g n P G going to infinity in PGL N `1pCq: in the KAK decomposition g n " k n a n k 1 n , the diagonal part a n goes to 8. Then, any probability measure on P N pCq which is invariant under all g n is supported in a proper projective subspace of P N pCq, and this contradicts our preliminary remark. So, G is compact in that case.

Step 2.-Now, assume that τ pGq is infinite. Identifying AutpA X q ˝with A X , τ pAutpXq ˝q is a complex algebraic subgroup of the torus A X , of positive dimension since it contains τ pGq. If the kernel of τ is finite, then AutpXq ˝is compact and virtually Abelian; thus, we may assume dimpkerpτ qq ě 1. In particular the fibers of α X have positive dimension, dimpα X pXqq ď 1 and α X pXq is a curve, which is elliptic because it is invariant under the action of τ pAutpXq ˝q. Then, the universal property of the Albanese morphism implies α X pXq " A X . In particular, α X is a submersion, for its critical values form a proper, τ pAutpXq ˝q-invariant subset of A X . Thus, X is a P 1 pCq-bundle over A X because the fibers of α X are smooth, are invariant under the action of kerpτ q, and can not be elliptic since otherwise X would be a torus. From [94, Thm 3] (see also [START_REF] Loray | Projective structures and projective bundles over compact Riemann surfaces[END_REF][START_REF] Potters | On almost homogeneous compact complex analytic surfaces[END_REF] for instance), there are two cases:

(1) either X " A X ˆP1 pCq, AutpXq " AutpA X q ˆPGL 2 pCq and we deduce as in the first step that G is a compact group;

(2) or AutpXq ˝is Abelian.

In both cases stiffness follows, and we are done.

Remark 10.6. Pushing the analysis further, it can be shown that, under the assumptions Proposition 10.5, Γ is relatively compact. Indeed in the last considered case, if Γ is not bounded it can be deduced from [94, Thm 3] that there are elements with wandering dynamics: all orbits in some Zariski open subset converge towards a section of α X . This contradicts the invariance of µ. Example 10.7 (See also [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF]). Consider an elliptic curve E " C{Λ and the Abelian surface A " E ˆE. The group GL 2 pZq determines a non-elementary group of automorphisms of E ˆE of the form px, yq Þ Ñ pax `by, cx `dyq. The involution η " ´id generates a central subgroup of GL 2 pZq, hence PGL 2 pZq acts on the (singular) Kummer surface A{η. Each singularity gives rise to a smooth P 1 pCq in the minimal resolution X of A{η, the group tB P PGL 2 pZq ; B " id mod 2u preserves each of these 16 rational curves, and its action on these curves is given by the usual linear projective action of PGL 2 pZq on P 1 pCq. In particular, it is proximal and strongly irreducible so it admits a unique, non-invariant, stationary measure.

The next result shows that when ν is symmetric, every non-invariant stationary measure is similar to the previous example.

Proposition 10.8. Let pX, νq be a random holomorphic dynamical system, with ν symmetric. Let µ be an ergodic ν-stationary measure giving positive mass to some proper Zariski closed subset of X. Then µ is supported on a Γ ν -invariant proper Zariski closed subset and (a) either µ is invariant; (b) or the Zariski closure of Supppµq is a finite, disjoint union of smooth rational curves C i , the stabilizer of C i in Γ induces a strongly irreducible and proximal subgroup of AutpC i q » PGL 2 pCq, and µpC i q ´1µ| C i is the unique stationary measure of this group of Möbius transformations.

Moreover, if pX, νq is non-elementary, the curves C i have negative self-intersection and can be contracted on cyclic quotient singularities.

Note that no moment assumption is assumed here. Before giving the proof, let us briefly discuss the question of stiffness for Möbius actions on P 1 pCq. Let ν be a symmetric measure on PGL 2 pCq. As already said, by Furstenberg's theory, if Γ ν is strongly irreducible and unbounded it admits a unique stationary measure, and this measure is not invariant. Otherwise, any νstationary measure is invariant because -either Γ ν is relatively compact and stiffness follows from [67, Thm. 3.5]; -or Γ ν admits an invariant set made of two points, then Γ ν is virtually Abelian and stiffness follows from Theorem 10.1; -or Γ ν is conjugate to a subgroup of the affine group AffpCq with no fixed point.

In the latter case after conjugating Γ ν to a subgroup of AffpCq we can write any g P Γ ν as gpzq " apgqz `bpgq. If apgq " 1 then Γ ν is Abelian and we are done. Otherwise Γ ν is merely solvable and we apply the following lemma which follows from a result of Bougerol and Picard (see [START_REF] Bougerol | Strict stationarity of generalized autoregressive processes[END_REF]Thm. 2.4]; a self-contained proof is provided in [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]). Lemma 10.9. Let ν be a symmetric probability measure on AffpCq. If no point of C is fixed by ν-almost every g, then the only ν-stationary probability on P 1 pCq is the point mass at 8.

Proof of Proposition 10.8. If µ has an atom then, by ergodicity, µ is supported on a finite orbit and it is invariant. So we now assume that µ is atomless. By ergodicity, µ gives full mass to a Γ ν -invariant curve D; let C 1 , . . . , C n be its irreducible components. Let Γ 1 be the finite index subgroup of Γ ν stabilizing each C i and ν 1 be the hitting measure induced by ν on Γ 1 ; it is symmetric, µ is ν 1 -stationary, and so are its restrictions µ| C i , for each C i .

If the genus of (the normalization of) C 1 is positive, then

Γ 1 | C 1 Ă AutpC 1 q is virtually Abelian, hence µ| C 1 is Γ 1 -invariant.
Since µ is ergodic, Γ ν permutes transitively the C i , and arguing as in Lemma 10.2, we see that µ is ν-invariant as well. Now, assume that the normalization Ĉ1 is isomorphic to P 1 pCq. If C 1 is not smooth, or if it intersects another Γ ν -periodic curve, then the image of Γ 1 in Autp Ĉ1 q » PGL 2 pCq is not strongly irreducible, and the discussion preceding this proof shows that µ is Γ 1 -invariant. Again, this implies that µ is Γ ν -invariant. The same holds if Γ 1 is a bounded subgroup of Autp Ĉ1 q. The only possibility left is that C 1 is smooth, disjoint from the other periodic curves, and Γ 1 induces a strongly irreducible subgroup of AutpC 1 q. Since Γ ν permutes transitively the C i , conjugating the dynamics of the groups Γ 1 | C i , the same property holds for each C i .

If Γ ν is non-elementary, Lemma 2.13 shows that C 2 i " ´m for some m ą 0, which does not depend on i because Γ ν permutes the C i transitively. Then, the C i being disjoint, one can contract them simultaneously, each of the contractions leading to a quotient singularity pC 2 , 0q{xηy with ηpx, yq " pαx, αyq for some root of unity α of order m (see [6, §III.5]). 10.5. Non-elementary groups: real dynamics. We now consider general non-elementary actions. As explained in the introduction, so far our results are restricted to subgroups of AutpXq preserving a totally real surface Y . We further assume that there exists a Γ ν -invariant volume form on Y ; this is automatically the case if X is an Abelian, a K3, or an Enriques surface (see [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF]). Note that, a posteriori, the results of §11 and [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF] suggest that measures supported on a totally real surface and invariant under a non-elementary subgroup of AutpXq tend to be absolutely continuous, unless they are supported by a curve or a finite set.

Theorem 10.10. Let pX, νq be a non-elementary random holomorphic dynamical system satisfying the moment condition (4.1). Assume that Y Ă X is a Γ ν -invariant totally real 2-dimensional smooth submanifold such that the action of Γ ν on Y preserves a probability measure vol Y equivalent to the Riemannian volume on Y . Then, every ergodic stationary measure µ on Y is:

(a) either almost surely invariant, (b) or supported on a Γ ν -invariant algebraic curve.

Since µ is not invariant, we are in case (2). Theorem 9.1 then implies that µ is supported on an invariant algebraic curve. This concludes the proof of the first assertions in Theorem 10.10, including the stiffness property when Γ has no periodic curve.

Step 2.-It remains to prove the last assertion. Let then µ be an ergodic stationary measure with h µ pX, νq ą 0. In the above trichotomy, ( 1) is now excluded. To exclude the alternative (2), by Theorem 9.1, it suffices to show that µ is not supported on an invariant curve. By Proposition 7.12 (i.e. the fibered Margulis-Ruelle inequality), µ is hyperbolic. If µ is supported on an algebraic curve, the proof of Corollary 8.3 leads to the following alternative: either µ is atomic or the Lyapunov exponent along that curve is negative. In the latter case µ is proximal along that curve and its stable conditionals are points. In both cases the fiber entropy would vanish, in contradiction with our hypothesis, so µ is not supported on an algebraic curve, as desired.

MEASURE RIGIDITY

In view of the results of Section 10 and [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF], it is natural to ask for a classification of invariant measures when Γ does not contain parabolic elements. If µ is a probability measure on X, we denote by Aut µ pXq the group of automorphisms of X preserving µ. If in addition the Lyapunov exponents of f with respect to µ satisfy λ ´pf, µq `λ`p f, µq ‰ 0, then case (a) does not occur, so Aut µ pXq is virtually cyclic. This result, and its proof, may be viewed as a counterpart, in our setting, to Theorems 5.1 and 5.3 of [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF]; again the possibility of invariant line fields is ruled out by using the complex structure. As before the typical case to keep in mind is when X is a projective surface defined over R and Y " XpRq. Observe that by ergodicity, if f preserves a smooth volume vol Y , then in case (a) µ will be the restriction of vol Y to an Aut µ pXq-invariant Borel set of positive volume.

Proof of Theorem 11.1. Since it admits a measure of positive entropy, f is a loxodromic transformation. By the Ruelle-Margulis inequality µ is hyperbolic with respect to f and it does not charge any point, nor any piecewise smooth curve: indeed, the entropy of a homeomorphism of the circle or the interval is equal to zero.

We first assume that X is projective; non-projective surfaces will be studied at the end of the proof. For µ-almost every x P X, the stable manifold W s pf, xq is an entire curve in X which is either transcendental or contained in a periodic rational curve (see [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF]Thm. 6.2]). Since f has only finitely many invariant algebraic curves (see [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF]Prop. 4.1]) and µ gives no mass to curves, W s pf, xq is µ-almost surely transcendental; then, the only Ahlfors-Nevanlinna current associated to W s pf, xq is T f ; similarly, the Ahlfors-Nevanlinna currents of the unstable manifolds give T f . (This is the analogue in deterministic dynamics of Theorem 8.2.) Fix g P Aut µ pXq and set Γ :" xf, gy. Our first goal is to prove the following:

Alternative: either Γ ˚is virtually cyclic and preserves tPrT f s, PrT f su Ă BH X ; or µ is absolutely continuous with respect to the Lebesgue measure on Y .

Let Y 1 Ă Y be the union of the connected components of Y of positive µ-measure. The measure µ does not charge any analytic subset of Y of dimension ď 1; thus, by analytic continuation, any h P Γ preserves Y 1 . So, without loss of generality we can replace Y by Y 1 .

We divide the argument into several cases according to the existence or non-existence of certain Γ-invariant line fields. In the first two cases we will conclude that Γ is elementary. In the third case, µ will be absolutely continuous with respect to the Lebesgue measure on Y ; then by the Pesin formula its Lyapunov exponents satisfy λ `pf, µq " ´λ´p f, µq " h µ pf q so when λ `pf, µq `λ´p f, µq ‰ 0, Case 3 is actually impossible.

Case 1.-There exists a Γ-invariant measurable line field. Specifically, we mean a measurable field of complex lines x Þ Ñ Epxq P PpT x Xq, defined on a set of full µ-measure, such that D x hpEpxqq " Ephpxqq for every h P Γ and almost every x P X; since µ is supported on the totally real surface Y , the field of real lines Epxq X T x Y Ă T x Y is also invariant, and determines Epxq. Now, µ being ergodic and hyperbolic for f , the Oseledets theorem shows that either Epxq " E s f pxq µ-almost everywhere or Epxq " E u f pxq µ-almost everywhere. Changing f into f ´1 if necessary, we may assume that Epxq " E s f pxq. Consider the automorphism h " g ´1f g P Aut µ pXq. Since h is conjugate to f , µ is also ergodic and hyperbolic for h. Thus, either E s h pxq " E s f pxq for µ-almost every x or E u h pxq " E s f pxq for µ-almost every x. Lemma 11.2. If there is a measurable set A of positive measure along which

E s h pxq " E s f pxq (resp. E u h pxq " E s f pxq), then W s pf, xq " W s ph, xq for almost every x in A (resp. W u ph, xq " W s pf, xq).
Let us postpone the proof of this lemma and conclude the argument. Suppose first that E s h pxq " E s f pxq on a subset A with µpAq ą 0. Then T f " T h because for µ-almost every x, the unique Ahlfors-Nevanlinna current associated to the (complex) stable manifold W s pf, xq (resp. W s ph, xq) is T f (resp. T h ). Since T h " Mpg ˚T f q ´1g ˚T f , we see that g, and therefore Γ itself, preserve the line RrT f s Ă H 1,1 pXq. Since rT f s 2 " 0, Γ fixes a point PrT f s of the boundary BH X , so it is elementary. Since in addition Γ contains a loxodromic element, Theorem 3.2 of [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF] shows that Γ ˚is virtually cyclic. Now, suppose that E u h pxq " E s f pxq on A. Then, T h " T f and the group generated by f and h is elementary. Since it contains a loxodromic element [27, Thm 3.2] says that xf ˚, h ˚y is virtually cyclic and fixes also PrT f s P BH X . This implies that g, hence Γ, preserves the pair of boundary points tPrT f s, PrT f su Ă BH X . Thus, in both cases Γ ˚is virtually cyclic and preserves tPrT f s, PrT f su Ă BH X .

Proof of Lemma 11.2. The argument is similar to that of Theorem 9.1, in a simplified setting, so we only sketch it. For µ-almost every x, W s pf, xq and W s ph, xq are tangent at x. Assume by contradiction that there exists a measurable subset A 1 of A of positive measure such that W s pf, xq ‰ W s ph, xq for every x P A 1 . Then for small ε ą 0 there exists two positive constants r " rpεq and c " cpεq, an integer k ě 2, and a measurable subset G ε Ă A 1 such that µpG ε q ą 0 and -W s loc pf, xq and W s loc ph, xq are well defined and of size r for every x P G ε , -W s loc pf, xq and W s loc ph, xq depend continuously on x on G ε Ă X, -inter x pW s loc pf, xq, W u loc pf, xqq " k for every x P G ε , -and osc pk,x,rq pW s r pf, xq, W s r ph, xqq ě c for every x P G ε . Indeed, to get the first and second properties, one intersects A 1 with a large Pesin set R ε . On A 1 X R ε the multiplicity of intersection x Þ Ñ inter x pW s loc pf, xq, W u loc pf, xqq is semi-continuous, so we can find k ě 2 and a subset R 1 ε Ă pA 1 X R ε q of positive measure such that (11.1) inter x pW s loc pf, xq, W u loc pf, xqq " k for every x P R 1 ε . Thus, the k-th osculation number is well defined, and the last property holds on a subset G ε Ă R 1 ε of positive measure if c is small. Let η s be a Pesin partition subordinate to the local stable manifolds of f . Since h µ pf q ą 0 the conditional measures µp¨|η s q are non-atomic. Thus there exists x P G ε such that x is an accumulation point of Supp `µp¨|η s pxqq| GεXη s pxq ˘. Fix a neighborhood N of x such that W s r pf, xq X W s r ph, xq X N " txu, and then pick a sequence px j q of points in G ε X η s pxq X N converging to x. The local stable manifolds W s r ph, x j q form a sequence of disks of size r at x j , each of them tangent to W s r pf, xq (at x j ), and all of them disjoint from W s r ph, xq (because x j does not belong to W s r ph, xq). This contradicts Corollary 9.9, and the proof is complete.

Case 2.-There is a pair of distinct measurable line fields tE 1 pxq, E 2 pxqu invariant under Γ. Again by the Oseledets theorem applied to f , necessarily tE 1 pxq, E 2 pxqu " tE s f pxq, E u f pxqu. For µ-almost every x, gptE s f pxq, E u f pxquq " tE s f pgpxqq, E u f pgpxqqu. As before, consider h " g ´1f g P Aut µ pXq. Since h is conjugate to f , it is hyperbolic and ergodic with respect to µ, and tE s f pxq, E u f pxqu " tE s h pxq, E u h pxqu for almost every x. Replacing h by h ´1 if necessary, there exists a set A of positive measure for which E s h pxq " E s f pxq, and we conclude as in Case 1. Case 3.-There is no Γ-invariant line field or pair of line fields. In other words, Cases 1 or 2 are now excluded. This part of the argument is identical to the proof of [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF]Thm 5.1.a].

First, we claim that there exists g 1 , g 2 P Γ and a subset A of positive measure such that D x g 1 pE s f pxqq R tE s f pg 1 pxqq, E u f pg 1 pxqqu and D x g 2 pE u f pxqq R tE s f pg 2 pxqq, E u f pg 2 pxqu for every x in A. Indeed since we are not in Case 2 (possibly switching E u f and E s f ) there exists g 1 P Γ and a set A of positive measure such that for x P A, D x g 1 pE s f pxqq Ć E s f pg 1 pxqq Y E u f pg 1 pxqq. Since we are not in Case 1, there exists g P Γ and a set B of positive measure such that for x P B, D x gpE u f pxqq ‰ E u f pgpxqq. If D x gpE s f pxqq P tE s f pgpxqq, E u f pgpxqqu on a subset B 1 of B of positive measure, then choose k ą 0 and ą 0 such that µpf pAq X B 1 q ą 0 and µpf k pgpf pAqqq X Aq ą 0 and define g 2 " g 1 f k gf ; otherwise, set g 2 " gf with such that µpf pAq X Bq ą 0. Then change A into A " A X f ´ pB 1 q (resp. A X f ´ pBq).

Denote by ∆ the simplex pa, b, c, dq P pR ˚q4 ; a `b `c `d " 1 ( . For α " pa, b, c, dq in ∆, let ν α be the probability measure ν α " aδ f `bδ f ´1 `cδ g `dδ g ´1 . Then µ is ν α -stationary and since µ is f -ergodic and ν α ptf uq ą 0, it is also ergodic as a ν α -stationary measure (see [12, §2.1.3]). Since we are not in Cases 1 or 2 and µ is hyperbolic for f , the invariance principle of Ledrappier [START_REF] Ledrappier | Positivity of the exponent for stationary sequences of matrices[END_REF] implies that the Lyapunov exponents of µ, viewed as a ν α -stationary measure, satisfy λ ά pµq ă λ ὰ pµq. (see Section 13.2.2 of [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF] and also [START_REF] Barrientos | Extremal exponents of random products of conservative diffeomorphisms[END_REF]; more precise statements and proofs can be found in [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]Thms. 11.10 and 11.11], and also [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces: hyperbolicity[END_REF])

Lemma 11.3. There exists a choice of α P ∆ such that µ is a hyperbolic ν α -stationary measure, i.e. λ ά pµq ă 0 ă λ ὰ pµq Proof. This is automatic when f and g are volume preserving because λ ά pµq " ´λὰ pµq in that case. For completeness, let us copy the proof given in [21, §13.2.4]. The assumptions of Case 3 and the strict inequality λ ´pµq ă λ `pµq imply that (11.2) α P ∆ Þ Ñ pλ ά pµq, λ ὰ pµqq P R 2 is continuous (see [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF]Prop. 13.7] or [START_REF] Viana | Lectures on Lyapunov exponents[END_REF]Chap. 9]). Since λ ά pµq ă λ ὰ pµq for every α P ∆, one of λ ά and λ ὰ is non zero. Furthermore, µ being invariant, the involution pa, b, c, dq Þ Ñ pb, a, d, cq interchanges the Lyapunov exponents. It follows that P " tα P ∆, λ ὰ ą 0u and N " tα P ∆, λ ά ă 0u are non-empty open subsets of ∆ such that P Y N " ∆. The connectedness of ∆ implies P X N ‰ H, as was to be shown.

Fix α P ∆ such that µ is hyperbolic as a ν α -stationary measure. The assumptions of Case 3 imply that the stable directions depend on the itinerary so the main result of [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF] shows that µ is fiberwise SRB (on the surface Y ), that is, the unstable conditionals of the measures µ x (here µ x " µ) are given by the Lebesgue measure (in some natural affine parametrizations of the unstable manifolds by the real line R). Since µ is invariant, we can revert the stable and unstable directions by applying the argument to F ´1, and we conclude that the stable conditionals are given by the Lebesgue measure as well. The absolute continuity property of the stable and unstable laminations then implies that µ is absolutely continuous with respect to the Lebesgue measure on Y .

Conclusion.-Assume that µ is not absolutely continuous with respect to the Lebesgue measure on Y . The above alternative holds for all subgroups Γ " xf, gy, with g P Aut µ pXq arbitrary. Therefore, if X is projective, then Aut µ pXq ˚preserves tPrT f s, PrT f su Ă BH X , which implies that Aut µ pXq ˚is virtually cyclic. By Lemma 3.18, Aut µ pXq ˚is also virtually cyclic when X is not projective. It remains to prove that Aut µ pXq itself is virtually cyclic. If not, then AutpXq is infinite, X is a torus C 2 {Λ (see Proposition 3.16), and Aut µ pXq X AutpXq ˝is a normal subgroup of Aut µ pXq containing infinitely many translations. This group is a closed subgroup of the compact Lie group AutpXq ˝" C 2 {Λ; thus, its connected component of the identity is a (real) torus H Ă C 2 {Λ of positive dimension. This torus H is invariant under the action of f by conjugacy. Since X " C 2 {Λ, f is a complex linear Anosov diffeomorphism of X, and it follows that dim R pHq ě 2. Being H-invariant, µ is then absolutely continuous with respect to the Lebesgue measure of Y ; this contradiction completes the proof.

It is natural to expect that the positive entropy assumption in Theorem 11.1 could be replaced by a much weaker assumption, namely, "µ gives no mass to proper Zariski closed subsets". We are able to deal with the following special case, which shows that the stiffness Theorem 10.10 takes a particularly strong form when Supppνq contains a Kummer example Theorem 11.4. Let f be a Kummer example on a compact Kähler surface X. Let µ be an atomless, f -invariant, and ergodic probability measure that is supported on a totally real, real analytic surface Y Ă X. If g P AutpXq preserves µ, then: Proof. Let us start with a preliminary remark. Assume that µpCq ą 0 for some irreducible curve C Ă X; since µ does not charge any point the support of µ |C is Zariski dense in C, and C is an f -periodic curve. But f being a Kummer example, such a curve is a rational curve C » P 1 pCq (obtained by blowing-up a periodic point of a linear Anosov map on a torus), on which f has a north-south dynamics; thus, all f -invariant probability measures on C are atomic, and we get a contradiction. This means that the assumption "µ has no atom" is equivalent to the assumption "µ gives no mass to proper Zariski closed subsets of X". Now, we follow step by step the proof of Theorem 11.1, only insisting on the required modifications. Since µ does not charge any curve, we can contract all f -periodic curves, and lift pf, µq to p f , μq, where f is a linear Anosov diffeomorphism of some compact torus C 2 {Λ and μ is an f -invariant probability measure (see [START_REF] Cantat | Holomorphic actions, Kummer examples, and Zimmer program[END_REF] for details on Kummer examples). We deduce that μ is hyperbolic for f and then, coming back to X, that µ is hyperbolic for f . Case 3 of the proof of Theorem 11.1 only requires hyperbolicity of µ so it carries over without modification. In Cases 1 and 2 we have to show that if Γ " xf, gy preserves a measurable line field or a pair of measurable line fields then Γ ˚is elementary. In either case we consider h " gf g ´1 and up to possibly replacing E u f by E s f and h by h ´1, we have E s f pxq " E s h pxq on a set of positive measure. But now f and h are Kummer examples so their respective stable foliations F s f and F s h are (singular) holomorphic foliations. From the previous reasoning F s f and F s h are tangent on a set of positive µ-measure; thus, F s f " F s h because the support of µ is Zariski dense. Moreover, every leaf of this foliation, except a finite number of algebraic leaves, is parametrized by C and the Ahlfors-Nevanlinna currents of these entire curves are all equal to the unique closed positive current T f that satisfies MpT f q " 1 and f ˚T f " λpf qT f . This implies that g ˚preserves R `rT f s or permute it with R `rT f s. Thus, a subgroup Γ 0 Ă Γ of index ď 2 preserves R `rT f s and by [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF]Thm. 3.2], Γ is virtually cyclic.

We expect that most results in this paper can be extended to polynomial automorphisms of R 2 . This is indeed the case for Theorem 11.1, with essentially the same proof.

Theorem 11.5. Let f be a polynomial automorphism of R 2 . Let µ be an ergodic f -invariant measure with positive entropy supported on R 2 . If g P AutpR 2 q satisfies g ˚µ " µ, then:

(a) either f and g are conservative and µ is the restriction of Leb R 2 to a Borel set of positive measure invariant under f and g;

(b) or the group generated by f and g is solvable and virtually cyclic; in particular, there exists pn, mq P Z 2 ztp0, 0qu such that f n " g m .

Remark 11.6. With the techniques developed in [START_REF] Cantat | Painlevé and Schrödinger[END_REF], the same result applies to the dynamics of OutpF 2 q acting on the real part of the character surfaces of the once punctured torus.

Proof. We briefly explain the modifications required to adapt the proof of Theorem 11.1, and leave the details to the reader. We freely use standard facts from the dynamics of automorphisms of C 2 . Let f and g be as in the statement of the theorem, and set Γ " xf, gy. Since its entropy is positive, f is of Hénon type in the sense of [START_REF] Lamy | L'alternative de Tits pour AutrC 2 s[END_REF]: this means that f is conjugate to a composition of generalized Hénon maps, as in [START_REF] Friedland | Dynamical properties of plane polynomial automorphisms[END_REF]Thm. 2.6]. Thus, the support of µ is a compact subset of C 2 , because the basins of attraction of the line at infinity for f and f ´1 cover the complement of a compact set; moreover, as in Theorem 11.1, µ cannot charge any proper Zariski closed subset. Let γ be an arbitrary element of Γ; then h :" γ ´1f γ is also of Hénon type. We run through Cases 1, 2 and 3 as in the proof of Theorem 11.1. Case 3 is treated exactly in the same way as above and implies that µ is absolutely continuous. This in turn implies that the Jacobian of f , a constant Jacpf q P C ˚since f P AutpC 2 q, is equal to ˘1; and since µ is ergodic for f , it must be the restriction of Leb R 2 to some Γ-invariant subset. In Cases 1 and 2, arguing as before and keeping the same notation, we arrive at W s ph, xq " W s pf, xq or W u pf, xq on a set of positive measure. For a Hénon type automorphism of C 2 , the closure of any stable manifold is equal to the forward Julia set J `, and J `carries a unique positive closed current T `of mass 1 relative to the Fubini Study form in P 2 pCq (see [START_REF] Sibony | Dynamique des applications rationnelles de P k . In Dynamique et géométrie complexes[END_REF]). So we infer that T h " T f or T h " T f ; as a consequence, the Green functions of f and h satisfy G h " G f or G h " G f , respectively.

Automorphisms of C 2 act on the Bass-Serre tree of AutpC 2 q, each u P AutpC 2 q giving rise to an isometry u ˚of the tree. If u is of Hénon type, then u ˚is loxodromic; its axis Geopu ˚q is the unique u ˚-invariant geodesic, and u ˚acts as a translation along it. From [START_REF] Lamy | L'alternative de Tits pour AutrC 2 s[END_REF]Thm. 5.4], G h " G f implies Geoph ˚q " Geopf ˚q; changing f into f ´1, G h " G f gives Geoph ˚q " Geopf ´1 ˚q " Geopf ˚q because Geopf ´1 ˚q " Geopf ˚q. Since γ ˚Geopf ˚q " Geoph ˚q, we see that Γ preserves Geopf ˚q; so, all u P Γ of Hénon type satisfy Geopu ˚q " Geopf ˚q. From [START_REF] Lamy | L'alternative de Tits pour AutrC 2 s[END_REF]Prop. 4.10], we conclude that Γ is solvable and virtually cyclic.

APPENDIX A. GENERAL COMPACT COMPLEX SURFACES

Here, we study the concept of non-elementary groups of automorphisms on (non Kähler) compact complex surfaces. We show that the two possible definitions of non-elementary group are equivalent and force the surface to be Kähler.

Let M be a compact manifold. We say that a group Γ of homeomorphisms of M is cohomologically non-elementary if its image Γ ˚in GLpH ˚pM ; Zqq contains a non-Abelian free subgroup, and that Γ is dynamically non-elementary if it contains a non-Abelian free group Γ 0 such that the topological entropy of every f P Γ 0 ztidu is positive. When M is a compact Kähler surface and Γ Ă AutpM q, Theorem 3.2 of [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF] and the fact that parabolic automorphisms have zero entropy imply that Γ is non-elementary (in the sense of Section 2.3.3) if and only if it is cohomologically non-elementary, if and only if it is dynamically non-elementary.

Lemma A.1. Let M be a compact manifold, and Γ be a subgroup of Diff 8 pM q. If Γ is cohomologically non-elementary, then Γ is dynamically non-elementary.

Proof. We split the proof in two steps, the first one concerning groups of matrices, and the second one concerning topological entropy.

Step 1.-Γ ˚contains a free subgroup Γ 1 , all of whose non-trivial elements have spectral radius larger than 1.

The proof uses basic ideas involved in Tits's alternative, here in the simple case of subgroups of GL n pZq. Let N be the rank of H t.f. pM ; Zq, where t.f. stands for "torsion free". Fix a basis of this free Z-module. Then Γ ˚determines a subgroup of GL N pZq. Our assumption implies that the derived subgroup of Γ ˚contains a non-Abelian free group Γ 0 of rank 2.

If all (complex) eigenvalues of all elements of Γ 0 have modulus ď 1, then by Kronecker's lemma all of them are roots of unity. This implies that Γ 0 contains a finite index nilpotent subgroup (see Proposition 2.2 and Corollary 2.4 of [START_REF] Benoist | Sous-groupes discrets des groupes de Lie[END_REF]), contradicting the existence of a non-Abelian free subgroup. Thus, there is an element f ˚in Γ 0 with a complex eigenvalue of modulus α ą 1. Let m be the number of eigenvalues of f ˚of modulus α, counted with multiplicities. Consider the linear representation of Γ 0 on Ź m H ˚pM ; Cq; the action of f ˚on this space has a unique dominant eigenvalue, of modulus α m ; the corresponding eigenline determines an attracting fixed point for f ˚in the projective space Pp Ź m H ˚pM ; Cqq; the action of f ˚on this topological space is proximal. Let

(A.1) t0u " W 0 Ă W 1 Ă ¨¨¨Ă W k Ă W k`1 " m ľ
H ˚pM ; Cq be a Jordan-Hölder sequence for the representation of Γ ˚: the subspaces W i are invariant, and the induced representation of Γ ˚on W i`1 {W i is irreducible for all 0 ď i ď k. Let V be the quotient space W i`1 {W i in which the eigenvalue of f ˚of modulus α m appears. Since Γ 0 is contained in the derived subgroup of Γ, the linear transformation of V induced by f ˚has determinant 1; thus, dimpV q ě 2. Now, we can apply Lemma 3.9 of [START_REF] Benoist | Sous-groupes discrets des groupes de Lie[END_REF] to (a finite index, Zariski connected subgroup of) Γ 0 | V : changing f is necessary, both f ˚|V and pf ´1q ˚|V are proximal, and there is an element g ˚in Γ ˚that maps the attracting fixed points a f and a f P PpV q of f ˚|V and pf ˚|V q ´1 to two distinct points (i.e. ta f , a f u X tgpa f q, gpa f qu " H) ; then, by the ping-pong lemma, large powers of f ˚and g ˚˝f ˚˝pg ˚q´1 generate a non-Abelian free group Γ 1 Ă Γ such that each element h ˚P Γ 1 ztidu has an attracting fixed point in PpV q. This implies that every element of Γ 1 ztidu has an eigenvalue of modulus ą 1 in H ˚pM ; Cq.

Step 2.-Since Γ 1 is free, there is a free subgroup Γ 1 Ă Γ such that the homomorphism Γ 1 Þ Ñ Γ 1 is an isomorphism. By Yomdin's theorem [START_REF] Yomdin | Volume growth and entropy[END_REF], all elements of Γ 1 ztidu have positive entropy, and we are done.

Theorem A.2. Let M be a compact complex surface, and Γ be a subgroup of AutpM q. Then, Γ is cohomologically non-elementary if and only if it is dynamically non-elementary. If such a subgroup exists, then M is a projective surface.

Proof. Indeed it was shown in [START_REF] Cantat | Dynamique des automorphismes des surfaces projectives complexes[END_REF] that every compact complex surface possessing an automorphism of positive entropy is Kähler. Thus, the first assertion follows from Lemma A.1 and Theorem 3.2 of [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF], and the second one follows from Theorem E.

The counting argument is identical to [7, §7], except that we apply the Ahlfors theory of covering surfaces to a union of disks, not just one. For notational ease, set " k , r " r j and Q " Q j ; Q is a subdivision of S » r0, 1s 2 by squares of size 2 ´j . We decompose Q as a union of four non-overlapping subdivisions Q , " 1, 2, 3, 4; by this we mean that for each , the squares Q P Q have disjoint closures Q. Fix such an and let q " #Q " 4 j´1 . Applying Ahlfors' theorem to each of the disks constituting ∆ n and summing over these disks, we deduce that the number of good components N pQ q satisfies ( 7 ) (B.3)

N pQ q ě pq ´4q area P 1 p∆ n q ´h length P 1 pB∆ n q,

where area P 1 (resp. length P 1 ) is the area of the projection p∆ n q (resp. length of pB∆ n q), counted with multiplicity, and h is a constant that depends only on the geometry of Q . Dividing by area P 1 p∆ n q, using length P 1 pB∆ n q " oparea P Finally, summing from " 1 to 4, we see that, relative to ˚κP 1 , the mass lost by discarding the bad components of size r in T is of order Opr 2 q: this is precisely the required estimate.

Let us now justify the geometric intersection statement, following step by step the proof of [56, Thm. 4.2]: let S be a current with continuous normalized potential on X; we have to show that S ^Tr increases to S ^T as r decreases to 0. Again the result is local so we work near x, use the projections 1 and 2 , and keep notation as above. Given squares Q, Q 1 P Q and a real number λ ă 1, we denote by λQ the homothetic of Q of factor λ with respect to its center, and by CpQ, Q 1 q the cube ´1 1 pQq X ´1 2 pQ 1 q. Fix ε ą 0. We want to show that for r ď rpεq, the mass of pT ´Tr q ^S is smaller than ε. The first observation is that there exists λpεq P p0, 1q, independent of r, such that translating Q if necessary, the mass of T ^S concentrated in Ť Q,Q 1 CpQ, Q 1 qzCpλQ, λQ 1 q is smaller than ε{2 (see [START_REF] Dujardin | Sur l'intersection des courants laminaires[END_REF]Lem. 4.5]). Fix such a λ. It only remains to estimate the mass of pT ´Tr q ^S in Ť Q,Q 1 CpλQ, λQ 1 q. In such a cube CpλQ, λQ 1 q the argument presented in [56, pp. 123-124], based on an integration by parts, gives the estimate (B.5) ż CpλQ,λQ 1 q pT ´Tr q ^S ď Cpλqmodcpu S , rq 1 r 2 M `pT ´Tr q| CpQ,Q 1 q ˘, where modcpu S , rq is the modulus of continuity of the potential u S of S. To conclude, we sum over all squares Q, Q 1 and use the estimate M pT ´Tr q " Opr 2 q to get that (B.6) M ´pT ´Tr q| Ť Q,Q 1 CpλQ,λQ 1 q ¯ď Cωpu S , rq. This is smaller than ε{2 if r ď rpεq.

(6. 15 )

 15 Kpx, yq " O ˆlog |x ´y| |x ´y| 2 ˙and ∇Kpx, yq " O ˆlog |x ´y| |x ´y| 3

Lemma 7 . 3 .

 73 The conditional measure m ξ on X ξ satisfies ν Z -almost surely m ξ " lim nÑ`8 pf ´1 ˝¨¨¨˝f ´nq › µ " lim nÑ`8

1 n log ˇˇJac η f n ξ pxq ˇˇ" ż log ˇˇJac η f 1 ξ

 11 , the integrability condition and the ergodic theorem imply that, almost surely, lim nÑ8 pxq ˇˇdmpξ, xq (7.14) " ż log ˇˇJac η f 1 ω pxq ˇˇdm `pω, xq " ż log |Jac η f pxq| dµpxqdνpf q.

  .e m ξ " m ζ almost surely when ζ P Σ u loc pξq; -the conditional measures of m with respect to this partition satisfy (see Equation (7.3)) (9.1) mp ¨| F ´px qq " ν Z p ¨| Σ u loc pξqq ˆδx . The next lemma can be seen as a complex analytic version of [21, Lemma 9.9]. Lemma 9.4. Let k ě 1 be an integer. Exactly one of the following assertions holds: (a) for m-almost every x " pξ, xq and for mp ¨| F ´pξ, xqq-almost every η inter x pW s loc pξ, xq, W s loc pη, xqq ě k `1; (b) for m-almost every x " pξ, xq and for mp ¨| F ´pξ, xqq-almost every η inter x pW s loc pξ, xq, W s loc pη, xqq ď k.

5 ) 1 2 ˇˇpf 1 ´f2 q pkq p0q ˇpk ´1q! |z| k´1 ď ˇˇf 1 1 pzq ´f 1 2 pzq ˇˇď 3 2 ˇˇpf 1

 5221 ´f2 q pkq p0q ˇpk ´1q! |z| k´1 . (9.6) Indeed put g " f 1 ´f2 " ř měk g m z m . Assumptions (i) and (iii) give|gpzq| ď 2r on Dprq, and g pkq p0q ‰ 0. By the Cauchy estimates, |g n | ď 2r 1´n for all n ě 0. Then on Dpαrq we get

10. 4 .

 4 Invariant algebraic curves II. Let us start with an example.

Theorem 11 . 1 .

 111 Let f be an automorphism of a compact Kähler surface X, preserving a totally real and real analytic surface Y Ă X. Let µ be an ergodic f -invariant measure on Y with positive entropy. Then (a) either µ is absolutely continuous with respect to the Lebesgue measure on Y ; (b) or Aut µ pXq is virtually cyclic.

  (a) either µ is absolutely continuous with respect to vol Y ; (b) or xf, gy is virtually isomorphic to Z.

  Hodge decomposition. Denote by H ˚pX ; Rq the cohomology of X with coefficients in the ring R; we shall use R " Z, Q, R or C. The group AutpXq acts on H ˚pX ; Cq, preserving the image of H ˚pX ; Zq; AutpXq ˚will denote the image of AutpXq in GLpH 2 pX; Cqq.

	The
	Hodge decomposition
	(2.1)

2.1. Cohomology.

2.1.1.

  The dimension of NSpX; Rq is the Picard number ρpXq. 2.1.2. Norm of f ˚. Let |¨| be any norm on the vector space H ˚pX ; Cq. If L is a linear transformation of H ˚pX ; Cq we denote by }L} the associated operator nom and if W Ă H ˚pX ; Cq is an L-invariant subspace of H ˚pX ; Cq, we denote by }L} W the operator norm of L| W .

  fixed point in H m and a unique fixed point in BH m ; -γ is loxodromic if γ has no fixed point in H m and exactly two fixed points in BH m . A subgroup Γ of O 1,m pRq is non-elementary if it does not preserve any finite subset of H m Y BH m . Equivalently Γ is non-elementary if and only if it contains two loxodromic elements with disjoint fixed point sets.

  2.3.3.The hyperbolic space H X . Let X be a compact Kähler surface. By the Hodge index theorem, the intersection form on H 1,1 pX, Rq has signature p1, h 1,1 pXq ´1q. The hyperboloid X is by definition this connected component, which is thus a model of the hyperbolic space of dimension h 1,1 pXq ´1. We denote by d H the hyperbolic distance, which is defined as before by coshpd H pu, vqq " xu | vy. From Lemma 2.2 and Corollary 2.5 we see that if }¨} is any norm on H ˚pX, Cq, then }f ˚} -› › pf ˚q´1 › › -xrκ 0 s | f ˚rκ 0 sy (here κ 0 is the fixed Kähler form introduced in Section 2.2).

	u P H 1,1 pX, Rq, xu | uy	(	" 1
	has two connected components, one of which intersecting the Kähler cone. The hyperbolic
	space H

  1,1 pX; Rq `' H 1,1 pX; Rq 0 .

	From now on we set:
	(2.11)	Π Γ :" H 1,1 pX; Rq `" NSpX; Rq `.
	Denote by H 2 pX; Rq 0 the direct sum of H 1,1 pX; Rq 0 and of the real part of H 2,0 pX; Cq '
	H 0,2 pX; Cq; then
	(2.10)	H 2 pX; Rq " H 1,1 pX; Rq `' H 2 pX; Rq 0
	and Γ ˚|H 2 pX;Rq 0 is contained in a compact group (see Lemma 2.1). The Néron-Severi group is
	Γ-invariant, and since X is projective it contains a vector with positive self-intersection. Then
	Proposition 2.8 and Lemma 2.6 imply:
	Proposition 2.11. Let X be a compact Kähler surface and Γ be a non-elementary subgroup of
	AutpXq. Then H 1,1 pX; Rq `" NSpX; Rq `is a Minkowski space, and the action of Γ on this
	space is non-elementary and strongly irreducible.
	Since non-elementary groups of isometries of H m occur only for m ě 2, we get:
	Corollary 2.12. Under the assumptions of Proposition 2.11, the Picard number ρpXq is greater
	than or equal to 3. If equality holds then NSpX; Rq `" NSpX; Rq and the action of Γ on
	NSpX; Rq is strongly irreducible.

  1,1 pXq ´2.Proof. Consider the Γ ν -invariant decomposition Π Γν 'Π K Γν given by Proposition 2.11 and Equation(2.11). Since the intersection form is negative definite on Π K Γν , the groupΓ ν | Π K Γν vanish.The linear action of Γ ν on Π Γν is strongly irreducible and non-elementary, hence not relatively compact. Therefore Furstenberg's theorem asserts that λ H 1,1 ą 0 (see e.g.[START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF] Thm III.6.3] or[START_REF] Benoist | Random walks on reductive groups, volume 62 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF] Cor 4.32]), and the remaining properties of the Lyapunov spectrum on Π Γν follow from the KAK decomposition in O 1,m pRq, with 1 `m " dimpΠ Γν q (see Lemma 2.4). 5.3. If a P H 1,1 pX; Rq satisfies a 2 ą 0, for instance if a is a Kähler class, then

	Γν	is bounded
	and all Lyapunov exponents of Γ ν | Π K
	Lemma lim nÑ`8

  Kpx, yq ^pf ˚κpyq ´Θpf ˚κqpyqq (in the notation of[START_REF] Dinh | Green currents for holomorphic automorphisms of compact Kähler manifolds[END_REF], f ˚κ and Θpf ˚κq correspond to Ω `and Ω ´respectively). The coefficients of the smooth p1, 1q-forms f ˚κ and Θpf ˚κq have their uniform norms bounded by C}f } 2 C 1 , where C " Cpκq ď C 1 }κ} 8 . The first estimate in (6.15) implies that the coefficients of K belong to L p loc for p ă 2, so it follows from the Hölder inequality that }φ} C 0 ď C 2 }κ} 8 }f } 2

		Prop. 2.1] that a solution to
	Equation (6.14) is given by
		ż
	(6.16)	φpxq "
		yPX
		C 1

  1,1 pX; Rq. For a P LimpΓq, let us set diam paq " Diampaq, where a is the unique pseudo-effective class of mass 1 such that Ppaq " a; this defines a measurable function on LimpΓq, by Lemma 6.4. Our purpose is to show that diam paq " 0 for µ B -almost every a. The stationarity of µ ż diam ´f n ¨¨¨f 1 paq ¯dνpf 1 q ¨¨¨dνpf n qdµ B paq (notice the order of compositions chosen here). Since the diameter is upper-semicontinuous it is uniformly bounded on LimpΓq. So, if we prove that diam `f n ¨¨¨f 1 paq ˘" 0 for ν N -almost every pf n q and every a, then we can apply the dominated convergence theorem to infer that diam paq " 0 µ B -almost surely. To derive the convergence (6.19), note that (6.20) diam ´f n ¨¨¨f 1 paq ¯" Diam pf n ¨¨¨f 1 aq M pf n ¨¨¨f 1 aq because Diam is homogeneous. Applying Lemma 6.8 and the multiplicativity of the Jacobian we get that (6.21) diam ´f n ¨¨¨f 1 paq ¯ď C log

	and iterating this relation gives
	ż
	(6.18)	diam paq dµ B paq "
	(6.19)	lim nÑ`8

B reads (6.17)

ż diam paq dµ B paq " żż diam `f ˚paq ˘dνpf qdµ B paq

  7.2.3. Hyperbolicity. It can happen that λ ´and λ `have the same sign. If λ ´and λ `are both negative, the conditional measures m ξ are atomic: this can be shown by adapting a classical Pesin-theoretic argument (see e.g. [78, Cor. S.5.2]) to the fibered dynamics of F on X (see[START_REF] Le | Équilibre statistique pour les produits de difféomorphismes aléatoires indépendants[END_REF] Prop. 2] for a direct proof and an example where the m ξ have several atoms). Such random dynamical systems are called proximal. For instance, generic random products of automorphisms of P 2 pCq, that is of matrices in PGLp3, Cq, are proximal; in such examples the stationary measure is not invariant. Other examples are given by contracting iterated function systems.When λ `and λ ´are both non-negative, we have the so-called invariance principle:Theorem 7.4. Let pX, νq be a random holomorphic dynamical system satisfying the integrability condition (4.1), and let µ be an ergodic stationary measure. If λ `pµq ě λ ´pµq ě 0 then µ is almost surely invariant.

	This result was proven by Crauel, building on ideas of Ledrappier *(see Theorem 5.1, Corol-lary 5.3 and Remark 5.6 in	SC: Ici j'ai juste enlevé référence interne à l'ancienn
		section 11

  Since f 1 ´f3 does not vanish and |f 1 ´f3 | ď δ ă 1 in Dprq, the function log |f 1 ´f3 | is harmonic and negative there. Thus for α ď 1{2, the Harnack inequality can be applied to ζ Þ Ñ pf 1 ´f3 qprζq in D: this gives (9.8). Likewise, we infer that

	(9.9)					1 3 pzq ˇˇď	1 αr	δ	1´2α 1`2α 0	.
	For this, recall the Harnack inequality: for any negative harmonic function in D
	(9.10)		1 ´|ζ| 1 `|ζ|	ď	upζq up0q	ď	1 `|ζ| 1 ´|ζ|	.
			1`2α					1´2α
	(9.11)	δ	1´2α 0	ď |f 1 pzq ´f3 pzq| ď δ	1`2α 0

  Remark 9.8. Lemma 9.7 does not hold in the real analytic setting. Indeed, take an integer n " 1 mod r4s and consider the n-th Chebychev polynomial T n , defined by T n pcos θq " cospnθq; it satisfies |T n | ď 1 on r´1, 1s, |T 1 n | ď 2n on r´1{2, 1{2s, and T 1 n p0q " n. Then, set P n pxq " 10 n 2 T `x ´5 n

	˘`25
	n 2

  3.-We now prove the almost sure invariance. As in [21, Eq. (11.1)] we consider a measurable partition P of X with the property that for m-almost every pξ, xq, Ppξ, xq Ă Σ s loc pξq ˆW s pξ, xq.

	By contradiction, assume that (9.23) fails. By contraction along the stable leaves, it follows that
	almost surely Σ s loc pξq ˆtxu is contained in
	(9.24)	Supp ´mp¨|Ppξ, xqq |Ppξ,xqzΣ s loc pξqˆtxu
	(9.22) rpξ,xq pξ, xq Ă The existence of such a partition is guaranteed, for instance, by Lemma 7.13. By [21, Prop Σ s loc pξq ˆW s
	11.1]( 5 ), to show that µ is almost surely invariant it is enough to prove that:
	(9.23)	for m almost every ξ, mp ¨|Ppξ, xqq is concentrated on Σ s loc pξq ˆtxu .

  10.3. Elementary groups. Recall that AutpXq is a topological group for the topology of uniform convergence and is in fact a complex Lie group (with possibly infinitely many connected components). Let AutpXq ˝be the connected component of the identity in AutpXq and Note that stiffness can fail when Γ ν is finite: see Example 10.4 below. The proof relies on the classification of elementary subgroups of AutpXq (see[START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF] Thm 3.2],

	(10.3)	AutpXq # " AutpXq{AutpXq	˝.

Let ρ : AutpXq Ñ GLpH ˚pX ; Zqq be the natural homomorphism; its image is AutpXq ˚" ρpAutpXqq (see § 2.1.1); is kernel contains AutpXq ˝and a theorem of Lieberman

[START_REF] David | Compactness of the Chow scheme: applications to automorphisms and deformations of Kähler manifolds[END_REF]

shows that AutpXq ˝has finite index in kerpρq. If Γ is a subgroup of AutpXq, we set Γ ˚" ρpΓq.

Theorem 10.3. Let X be a compact Kähler surface. Let ν be a symmetric probability measure on AutpXq satisfying the moment condition (4.1). If Γ ν is elementary and Γ ν is infinite, then pX, νq is stiff.

  1 p∆ n qq, which is guaranteed by Ahlfors' construction, and letting n go to `8, we obtain (B.4) xT Q | Q , ˚κP 1 y ě pq ´4qr 2 " area P 1 ´ďSPQ S ¯´4r 2 .

  SERGE CANTAT, IRMAR, CAMPUS DE BEAULIEU, B ÂTIMENTS 22-23 263 AVENUE DU G ÉN ÉRAL LECLERC, CS 74205 35042 RENNES C ÉDEX Email address: serge.cantat@univ-rennes1.fr SORBONNE UNIVERSIT É, CNRS, LABORATOIRE DE PROBABILIT ÉS, STATISTIQUE ET MOD ÉLISATION (LPSM), F-75005 PARIS, FRANCE Email address: romain.dujardin@sorbonne-universite.fr

This actually requires checking that the whole proof of[START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF] can be reproduced in our complex setting: we will come back to this issue in a forthcoming paper. Since we are just using this remark here in Corollary 9.2 we take the liberty to anticipate on that research.

Brown and Rodriguez-Hertz make it clear that this result holds for an arbitrary smooth random dynamical system on a compact manifold.

The proof in[START_REF] Raugi | A general Choquet-Deny theorem for nilpotent groups[END_REF] is not correct (Lemma 2.5 there is false) but it works perfectly, and is quite short, if the support of ν is countable or if the nilpotency class is ď 2. See the introduction of[START_REF] Raugi | A general Choquet-Deny theorem for nilpotent groups[END_REF] for a summary of previous results.

Note: A longer version of this paper is maintained on the personal website of the authors (see [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]). It contains complementary examples and preliminaries, and the proofs of some known facts are included for convenience.

APPENDIX B. STRONG LAMINARITY OF AHLFORS CURRENTS

In this appendix, we sketch the proof of Lemma 8.8, explaining how to adapt arguments of [7,[START_REF] Dujardin | Laminar currents in P 2[END_REF][START_REF] Dujardin | Sur l'intersection des courants laminaires[END_REF], written for X " P 2 pCq, to our context.

Proof of Lemma 8.8. Let p∆ n q be a sequence of unions of disks, as in the definition of injective Ahlfors currents, such that 1 Mp∆nq t∆ n u converges to T . Since X is projective we can choose a finite family of meromorphic fibrations i : X P 1 such that -the general fibers of i are smooth curves of genus ě 2; -for every x P X, there are at least two of the fibrations i , denoted for simplicity by [START_REF] Avila | Extremal Lyapunov exponents: an invariance principle and applications[END_REF] and 2 , which are well defined in some neighborhood U x of x (x is not a base point of the corresponding pencils), satisfy pd 1 ^d 2 qpxq ‰ 0 (the fibrations are transverse), and for which the fibers ´1 k p k pxqq containing x are smooth. If we blow-up the base points of k , k " 1, 2, we obtain a new surface X 1 Ñ X on which each k lifts to a regular fibration 1 k ; the open neighborhood U x is isomorphic to its preimage in X 1 so, when working on U x , we can do as if the two fibrations k were local submersions with smooth fibers of genus ě 2.

To construct T r , we follow the proof of [56, Proposition 4.4] (see also [START_REF] Dujardin | Laminar currents in P 2[END_REF]Proposition 3.4]). The construction will work as follows: we fix a sequence pr j q converging to zero, and for every j we extract from 1 Mp∆nq t∆ n u a current T n,r j made of disks of size « r j which are obtained from ∆ n by only keeping graphs of size r j over one of the projections i .

By a covering argument, it is enough to work locally near a point x, with two projections 1 and 2 as above. Let S Ă C be the unit square tx `iy ; 0 ď x ď 1, 0 ď y ď 1u » r0, 1s 2 . To simplify the exposition, we may assume that (B.1) k pU x q " S Ă C Ă P 1 pCq pfor k " 1, 2q. Set r j " 2 ´j and consider the subdivision Q j of S » r0, 1s 2 into 4 j squares Q of size r j . A connected component of ∆ n X ´1 k pQq, for such a small square Q, is called a graph (with respect to k ) if it lifts to a local section of the fibration 1 k : X 1 Ñ P 1 pCq above Q. Then, we fix j, intersect ∆ n with ´1 k pQq, and keep only the components of ´1 k pQ X ∆ n q, Q P Q j which are graphs with respect to k . Such a family of graphs is normal because the fibers of 1 k have genus ě 2 (compare to Lemma 3.5 of [START_REF] Dujardin | Laminar currents in P 2[END_REF]).

This being done, we can copy the proof of [START_REF] Dujardin | Sur l'intersection des courants laminaires[END_REF]Proposition 4.4]. Letting n go to `8 and extracting a converging subsequence, we obtain a uniformly laminar current T Q j ,k ď T . Away from the base points of k , T Q j ,k is made of disks of sizer j which are limits of disks contained in the ∆ n . Combining the two currents T Q j ,k , we get a current T r j ď T which is uniformly laminar in every cube ´1 1 pQq X ´1 2 pQ 1 q, Q, Q 1 P Q j , and such that (B.2)

where κ P 1 is the Fubini-Study form. By definition, T will be strongly approximable if locally MpT ´Tr j q ď Opr 2 j q. Using the fact that 1 κ P 1 ` 1 κ P 1 ě Cκ 0 and the Inequality (B.2), it will be enough to show that xT ´TQ j ,k , k κ P 1 y " Opr 2 j q for k " 1, 2. This itself reduces to counting (with multiplicity) the number of "good components" of ∆ n for the projections k : ∆ n Ñ Q j that is, the components above the squares Q of Q j that are kept in the above contruction of T Q j ,k (the graphs relative to k ).