IRMAR Serge Cantat
email: serge.cantat@univ-rennes1.fr

Romain Dujardin
email: romain.dujardin@sorbonne-universite.fr

HYPERBOLICITY FOR LARGE AUTOMORPHISM GROUPS OF PROJECTIVE SURFACES

We study the hyperbolicity properties of the action of a non-elementary automorphism group on a compact complex surface, with an emphasis on K3 and Enriques surfaces. A first result is that when such a group contains parabolic elements, Zariski diffuse invariant measures automatically have non-zero Lyapunov exponents. In combination with our previous work, this leads to simple criteria for a uniform expansion property on the whole surface, for groups with and without parabolic elements. This, in turn, has strong consequences on the dynamics: description of orbit closures, equidistribution, ergodicity properties, etc.

Along the way, we provide a reference discussion on uniform expansion of non-linear discrete group actions on compact (real) manifolds and the construction of Margulis functions under optimal moment conditions.

INTRODUCTION

This article is a follow-up to [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF], [START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF] and [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF]. Let X be a compact complex surface and let AutpXq denote its group of automorphisms, i.e. of holomorphic diffeomorphisms. Let Γ be a subgroup of AutpXq. We say that Γ is non-elementary if the subgroup Γ › ď GLpH › pX, Cqq induced by the action of Γ on the De Rham cohomology of X contains a non-abelian free group; the existence of a non-elementary subgroup of AutpXq implies that X is projective (see [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]Thm 3.17 and Thm A.1]). Our purpose in this series of papers is to study the dynamics of such a non-elementary group Γ on X, notably by means of random walk techniques.

1.1.

Wehler examples. To understand the motivation behind our general results, it is interesting to start with the Wehler family W of surfaces of degree p2, 2, 2q in P 1 ˆP1 ˆP1 , which has been a recurring example in our work (see e.g. [19, §3.1]). This family W depends on 26 parameters and is naturally parameterized by P 26 pCq; we shall denote by W 0 Ă W the Zariski open subset of smooth Wehler surfaces which do not contain any fiber of the three coordinate projections P 1 ˆP1 ˆP1 Ñ P 1 . For X P W 0 , the three natural projections X Ñ P 1 ˆP1 are ramified covers of degree 2; their deck transformations yield three holomorphic involutions σ 1 , σ 2 , and σ 3 ; the group Γ generated by these involutions is non-elementary and isomorphic to Z{2Z ˚Z{2Z ˚Z{2Z.

Since every X P W 0 is a K3 surface, there is a canonical AutpXq-invariant volume form vol X on XpCq; furthermore, when X is defined over R there is a canonical area form vol XpRq on XpRq which is invariant under the action of AutpX R q (see Example 1.4 below). Slightly abusing notation, we respectively denote by vol X and vol XpRq the associated measures on X and XpRq, normalized to have mass 1.

Our first main result is a complete description of orbit closures for most parameters X P W 0 . Recall that a 2-dimensional real submanifold Y Ă X is totally real if for every x P Y , T x Y spans T x X as a complex vector space.

Theorem 1.1. There exists a dense Zariski open subset W exp Ă W 0 pCq such that for every X P W exp , there exists a Γ-invariant finite set F Ă X and a Γ-invariant totally real analytic surface Y Ă X (with possibly finitely many singular points) with the following property: for every x P X, (a) either x P F (and its orbit is finite); (b) or Γpxq is a union of connected components of Y ; (c) or Γpxq " X.

In this statement both F and Y may be empty, depending on X. For instance, [18, Thm A] says that F is empty for a dense set of Wehler surfaces X P W 0 pCq. A typical situation for case (b) is that X is defined over R and Y " XpRq.

If we restrict to real parameters in W, we also have a fairly complete understanding of the asymptotic distribution of random orbits. By this we mean the following. Let ν be the probability measure on Γ defined by ν " 1 3 pδ σ 1 `δσ 2 `δσ 3 q. For any x in XpRq, and for any sequence pg i q of automorphisms g i P Γ chosen independently with distribution ν, consider the trajectory pg n ¨¨¨g 0 pxqq ně0 . Let X 1 pRq be a union of connected components of XpRq. We say that these random trajectories, starting at x, are equidistributed in X 1 pRq if for ν N -almost every pg i q, the empirical measures 1 n ř n´1 k"0 δ g k ¨¨¨g 0 pxq converge to the normalized volume form induced by vol XpRq on X 1 pRq as n Ñ 8. The appearance of X 1 pRq is due to the fact that Γ may not act transitively on the components of XpRq.

Theorem 1.2. There exists a dense Zariski open subset W exp pRq Ă W 0 pRq such that for every X P W exp pRq, there exists a Γ-invariant finite set F Ă XpRq such that for every x P XpRq: (a) either x P F ; (b) or the random trajectories starting at x are equidistributed in a union of connected components of XpRq.

An interesting point in Theorems 1.1 and 1.2 is that their conclusions hold for every x P X. Let us explain how these theorems fall within the progression of [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF][START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF][START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF] and what the last missing ingredient was until the present paper.

First, the existence of the maximal finite invariant set F follows from [START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF]Thm C]. One key point here is that Γ contains parabolic elements, that is automorphisms whose action on H ˚pX ; Cq is virtually unipotent and of infinite order (see Section 6). Now, the scheme of proof of Theorem 1.2 is as follows. The random walk on Γ induced by ν gives rise to a random dynamical system on X. We refer to [START_REF] Kifer | Ergodic theory of random transformations[END_REF][START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF] for general references on this topic, and to Sections 4 and 7 of [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF] for our holomorphic context. In particular, we shall use the notions of stationary and invariant measures µ, of fibered entropy h µ pX, νq, etc. Fix x P XzF . By Breiman's ergodic theorem, every cluster value of the sequence of empirical measures 1 n ř n´1 k"0 δ g k ¨¨¨g 0 pxq is a ν-stationary measure. We proved in [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF] that every ν-stationary measure is Γ-invariant, and in [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF] we showed that any invariant measure is either supported on F , or of the form vol X 1 pRq , for some union of components X 1 pRq of XpRq. Thus, any cluster value of 1 n ř n´1 k"0 δ g k ¨¨¨g 0 pxq is a convex combination of point masses and vol X 1 pRq . The last step is to show that if Γpxq is infinite, the limiting empirical measures give no mass to F .

For Theorem 1.1 the situation is similar: most of the work was done in [20, §8], except that there we could not exclude that the accumulation locus of an infinite orbit could be contained in a finite invariant set.

These difficulties were already addressed for homogeneous random dynamical systems in [START_REF] Benoist | Stationary measures and invariant subsets of homogeneous spaces (III)[END_REF][START_REF] Eskin | Random walks on locally homogeneous spaces[END_REF] and in the context of non-linear actions on real surfaces in [START_REF] Liu | Lyapunov exponents approximation, symplectic cocycle deformation and a large deviation theorem[END_REF][START_REF] Chung | Stationary measures and orbit closures of uniformly expanding random dynamical systems on surfaces[END_REF]. The key is to show that if X belongs to the dense open set W exp of Theorem 1.1 (resp. W exp pRq of Theorem 1.2), the maximal finite invariant set F is repelling for the random dynamics. Since we do not know the set F , nor its cardinality (examples of Wehler surfaces with large finite invariant sets were recently constructed in [START_REF] Fuchs | Orbits on K3 surfaces of Markoff type[END_REF]), we make a large detour and prove a uniform hyperbolicity property for the dynamics on the whole of X: this is the uniform expansion property that we present in detail in § 1.3. Establishing this property relies on ergodic-theoretic arguments, the first of which is an automatic hyperbolicity property that we describe in the next paragraph. 1.2. Hyperbolicity of invariant measures. It is a fundamental (and widely open) problem in conservative dynamics to show the typicality of non-zero Lyapunov exponents on a set of positive Lebesgue measure. In deterministic dynamics, a recent breakthrough is the work of Berger and Turaev [START_REF] Berger | On Herman's positive entropy conjecture[END_REF]. Adding some randomness makes such a hyperbolicity result easier to obtain: see [START_REF] Blumenthal | Lyapunov exponents for random perturbations of some area-preserving maps including the standard map[END_REF] for random perturbation of the standard map, and [START_REF] Barrientos | Extremal exponents of random products of conservative diffeomorphisms[END_REF][START_REF] Obata | Positive exponents for random products of conservative surface diffeomorphisms and some skew products[END_REF] for random conservative diffeomorphisms on closed real surfaces. The results of Barrientos and Malicet [START_REF] Barrientos | Extremal exponents of random products of conservative diffeomorphisms[END_REF] and of Obata and Poletti [START_REF] Obata | Positive exponents for random products of conservative surface diffeomorphisms and some skew products[END_REF] are perturbative in nature, so they do not give explicit examples. In our context, the rigidity properties of holomorphic diffeomorphisms will enable us to exhibit explicit criteria ensuring such a non-uniform hyperbolicity.

In [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF] we have classified invariant measures for non-elementary groups containing parabolic elements. We say that a measure µ on X is Zariski diffuse if it gives zero mass to proper Zariski closed subsets. If µ is Γ-invariant and ergodic for some Γ Ă AutpXq, this is equivalent to its support Supppµq being Zariski dense. Roughly speaking, our classification of invariant measures says that every Zariski diffuse, ergodic probability measure is given by an analytic 4form on X or by an analytic 2-form on some invariant, real analytic subset Y Ă X of dimension 2. Here we proceed to a finer study of the dynamical properties of these invariant measures. For this, we fix a probability measure ν on AutpXq, satisfying the moment condition (M)

ż ´log }f } C 1 pXq `log › › f ´1› › C 1 pXq ¯dνpf q ă `8,
and view any invariant measure µ as a ν-stationary measure, that is, ş f › µ dνpf q " µ. Then by (M), the Lyapunov exponents of µ are well defined. We denote by Γ ν Ă AutpXq the closed subgroup generated by Supppνq (1 ).

Theorem 1.3. Let X be a compact complex surface and Γ be a non-elementary subgroup of AutpXq containing parabolic elements. Let µ be a Zariski diffuse ergodic Γ-invariant probability measure on X. Let ν be any probability measure on AutpXq satisfying Γ ν " Γ and the moment condition (M).

Then, viewed as a ν-stationary measure, µ is hyperbolic and its fiber entropy h µ pX, νq is positive.

A variant of this result will also be obtained when Γ ν contains a Kummer example instead of a parabolic element (see Theorem 7.4).

Example 1.4. When X is a torus or a K3 surface, the canonical bundle K X is trivial and, up to multiplication by a complex number of modulus 1, there is a unique section Ω X of K X that satisfies ş X Ω X ^ΩX " 1. The volume form vol X :" Ω X ^ΩX is AutpXq-invariant. Likewise, every Enriques surface S inherits such an invariant volume form vol S from its universal cover X (a 2-to-1 cover by a K3 surface). Under the assumptions of Theorem 1.3, vol X is Γ-ergodic, thus we conclude that it is hyperbolic. Other examples are provided by some rational surfaces (see the discussion on Coble surfaces in [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]).

In these situations the 2-form Ω X also induces a natural measure vol Y on any totally real surface Y Ă X (see [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF]Rmk 2.3]). For instance, if X is projective and defined over R, Γ is contained in AutpX R q, and Y is a Γ-invariant connected component of XpRq, Theorem 1.3 asserts that vol Y is hyperbolic.

1.3. Uniform expansion. Fix a riemannian metric on X. We say that the measure ν on AutpXq is uniformly expanding if there exists c ą 0 and an integer n 0 such that for every x P X and every v P T x Xzt0u, (1.1) ż AutpXq log }D x f pvq} }v} dν pn 0 q pf q ě c; here ν pnq denotes the n th convolution power of ν. This notion is taken from [START_REF] Chung | Stationary measures and orbit closures of uniformly expanding random dynamical systems on surfaces[END_REF][START_REF] Dolgopyat | On simultaneous linearization of diffeomorphisms of the sphere[END_REF][START_REF] Liu | Lyapunov exponents approximation, symplectic cocycle deformation and a large deviation theorem[END_REF][START_REF] Zhang | On stable transitivity of finitely generated groups of volume-preserving diffeomorphisms[END_REF] (see also [START_REF] Eskin | Random walks on locally homogeneous spaces[END_REF][START_REF] Prohaska | Markov random walks on homogeneous spaces and Diophantine approximation on fractals[END_REF] for the linear context) and has a number of strong ergodic and topological consequences on the action of Γ ν . So far, uniform expansion has been verified only in the context of homogeneous dynamics, or for certain perturbative situations, or with the help of numerical methods. The geometric analysis of stationary measures developed in [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF] together with Theorem 1.3 will be used to obtain the following result.

Theorem 1.5. Let X be a compact complex surface which is not rational. Let ν be a probability measure on AutpXq. Assume that: (i) ν satisfies the moment condition (M) and (ii) the group Γ " Γ ν is non-elementary and contains parabolic elements. Then ν is uniformly expanding if and only if the following two conditions hold:

(1) every finite Γ-orbit is uniformly expanding;

(2) there is no Γ-invariant algebraic curve.

Here, by definition, a finite orbit F of Γ is said uniformly expanding if condition (1.1) holds for every x P F . This is the repulsion property alluded to at the end of § 1.1.

Checking condition [START_REF] Baouendi | Real submanifolds in complex space and their mappings[END_REF] is not hard in practice and boils down to cohomological computations (see §6.3). However, in a non-linear setting, Condition (1) is more delicate to verify. On the positive side, if X is not a torus and condition (2) holds, then by [START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF] there are only finitely many finite Γ-orbits. Moreover, if ν is symmetric and satisfies a slightly stronger moment condition (M `), Theorem 8.16 provides a checkable necessary and sufficient condition for a given finite Γ-orbit to be uniformly expanding: it is equivalent to the tangent action of Γ being proximal and strongly irreducible. It follows that when ν is symmetric (and X is not rational) the uniform expansion property depends only on Γ, and not on ν (Corollary 8.17).

On the negative side, so far there is no a priori bound on the number of finite orbits. Nevertheless, we show in Theorem 9.1 that under the assumptions of Theorem 1.5, if there is no invariant algebraic curve, there is a computable number N " N pX, Γq such that any finite orbit of length greater than N is uniformly expanding (see § 9.1 for details on what we mean by computable here). Consequently there is a simple algorithm for checking uniform expansion: verify that finite orbits of length at most N have non-elementary tangent action.

An immediate consequence of Theorem 1.5 is the following.

Corollary 1.6. Under the assumptions of Theorem 1.5, if there is no proper algebraic Γ νinvariant subset, then ν is uniformly expanding.

Since uniform expansion is an open property in the C 1 topology, and since by [18, Thm A] a dense set of Wehler examples has no proper Zariski closed invariant set, this implies that uniform expansion is satisfied on an open and dense set of Wehler examples. To deduce Theorems 1.1 and 1.2, we must further show that this open subset is Zariski open: for this, we use the effective result described above, and prove that in the Wehler family, the integer N pX, Γq is uniformly bounded (here Γ " xσ 1 , σ 2 , σ 3 y; see Theorem 9.3 and Proposition 9.7). In particular the equations defining W exp and W exp pRq could in principle be written down explicitly. 1.4. Ergodicity. Given an action of a general non-elementary group Γ on a compact complex surface X, one may ask the following two basic questions: does there exist a dense orbit? is the action ergodic with respect to Lebesgue measure? (The latter makes sense even when there is no invariant volume form.) If Γ contains a parabolic element, by [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF] the answer to both questions is 'yes', but without parabolic elements, the answer is unknown. A natural obstruction to the existence of a dense orbit could be the presence of a non-trivial Fatou component for Γ. No example of such a Fatou component is known so far; note that examples do exist for algebraic actions on affine surfaces (see [START_REF] Cantat | Painlevé and Schrödinger[END_REF][START_REF] Rebelo | Dynamics of groups of automorphisms of character varieties and Fatou/Julia decomposition for Painlevé[END_REF]).

As a matter of fact, the failure of ergodicity is associated to a lack of expansion: indeed a theorem of Dolgopyat and Krikorian [START_REF] Dolgopyat | On simultaneous linearization of diffeomorphisms of the sphere[END_REF] asserts that a conservative uniformly expanding action on a (real) surface must be ergodic. It is not difficult to extend their argument to the complex setting (see Theorem 10.2). In Theorem 8.9 we state a general criterion (i.e. without parabolic elements) for uniform expansion which shows that under the conditions (1) and ( 2) of Theorem 1.5, the failure of uniform expansion, is due to the existence of a Γ-invariant measure with exceptional properties (see Theorems 8.9 and B.1). We expect it to be an extremely rare phenomenon. Incidentally, this shows that the question of ergodicity for general non-elementary groups (i.e. without parabolic elements) ultimately boils down to the classification of Γ-invariant measures.

Another interesting consequence of our results, together with [START_REF] Dolgopyat | On simultaneous linearization of diffeomorphisms of the sphere[END_REF], is that a generic real Wehler example is stably ergodic among C 2 volume preserving actions, that is, if X belongs to the open set W exp pRq of Theorem 1.2 and σ 1 1 , σ 1 2 , σ 1 3 are C 2 volume preserving diffeomorphisms sufficiently close to of σ 1 , σ 2 , σ 3 in the C 1 topology, then Γ 1 :" xσ 1 1 , σ 1 2 , σ 1 3 y is ergodic for vol XpRq .

In an opposite direction, the examples from [20, §9] of AutpX R q-invariant domains with boundary in XpRq (which admit an invariant curve) provide explicit counterexamples to uniform expansion.

1.5. Organization of the paper. The first part of this paper (Sections 2 to 5) is devoted to a general study of the notion of uniform expansion on compact (real) manifolds. Much of this material is inspired from other sources; the novelty here is that we strive for optimal moment conditions. In Section 2 we give several equivalent definitions of uniform expansion: this is inspired by Liu [START_REF] Liu | Lyapunov exponents approximation, symplectic cocycle deformation and a large deviation theorem[END_REF] and Chung [START_REF] Chung | Stationary measures and orbit closures of uniformly expanding random dynamical systems on surfaces[END_REF]. In Section 3 we show that uniform expansion is preserved when restricting to a finite index subgroup or taking a finite extension (Proposition 3.3); this is useful when dealing with invariant sets made of finitely many connected components. Section 4 deals with the construction of Margulis functions. In a nutshell, a Margulis function near a finite uniformly expanding invariant set F is a function u : M zF Ñ R `that tends to infinity at F and decreases on average along orbits. The existence of such a function guarantees that empirical measures of random orbits do not accumulate at F . These functions have played an important role in random dynamics since the work of Eskin and Margulis [START_REF] Eskin | Recurrence properties of random walks on finite volume homogeneous manifolds[END_REF]. Here, thanks to the work by Bénard and De Saxcé [START_REF] Énard | Random walks with bounded first moment on finite-volume spaces[END_REF], we construct such Margulis functions under optimal moment conditions (Theorem 4.1); note that the usual average decay property ş upf pxqqdνpf q ď aupxq`b, a ă 1, is then replaced by ş upf pxqqdνpf q ď upxq´γ, γ ą 0. This repulsion property does not hold if F is an invariant submanifold (see Example 4.6). However in the holomorphic context, Margulis functions can be constructed for invariant totally real manifolds of maximal dimension (Theorem 4.5): a typical situation is that of XpRq Ă X for real projective manifolds. In Section 5, we elaborate on an ergodic-theoretic criterion for uniform expansion borrowed from [START_REF] Chung | Stationary measures and orbit closures of uniformly expanding random dynamical systems on surfaces[END_REF].

In the second part of the paper (Sections 6 to 10), we consider groups of automorphisms of projective surfaces. Theorem 1.3 is established in Section 7. In Section 8 we prove a general version of Theorem 1.5 and study uniform expansion along periodic orbits; this makes essential use of the results of the first part. The focus in § 9.1 is on finding algorithmically checkable conditions for uniform expansion along finite orbits (Theorem 9.1); this leads to a precise description of the locus of uniform expansion in the Wehler family (Theorem 9.3). In § 9.2, we construct uniformly expanding actions by perturbing Kummer examples in the Wehler family; in particular this work for "thin" subgroups of AutpXq containing no parabolic element. In Section 10 we study orbit closures and equidistribution by proving general versions of Theorems 1.1 and 1.2; we also explain the adaptation to the complex setting of the ergodicity theorem of Dolgopyat and Krikorian [START_REF] Dolgopyat | On simultaneous linearization of diffeomorphisms of the sphere[END_REF].

The paper ends with two appendices on the synchronization property and the rigidity of zero entropy measures. 1.6. Notes and comments. Theorem 1.3 was included in the first preprint version of [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]. We were informed of ongoing projects by Aaron Brown, Alex Eskin, Simion Filip and Federico Rodriguez Hertz, as well as Megan Roda, on the classification of stationary measures for uniformly expanding actions. This should fit nicely with our work; indeed, parts of this article are written so as to to be easily combined with such a classification (see e.g. Theorem 10.5),

We are grateful to Jean-Franc ¸ois Quint for useful comments on Margulis functions.

Part 1. Uniform expansion for discrete group actions on manifolds

GENERALITIES

In this section, M denotes a compact manifold and ν is a probability measure on the group Diff 1 pM q of C 1 diffeomorphisms of M . We fix a Riemannian metric on M . We denote by }¨} the norm induced by the metric on the tangent bundle T M , and by T 1 M the unit tangent bundle.

2.1. Moment conditions. If f is a C 1 -diffeomorphism of M , we denote by f › its action on T M . Note that if v P T M is a tangent vector based at x (that is, v P T x M ), then f › v " D x f pvq is based at f pxq. By definition, }f } C 1 pXq is the supremum of v Þ Ñ }f › v} on T 1 M . For f P Diff 1 pM q we put (2.1) Lpf q " log }f } C 1 pXq `log › › f ´1› › C 1 pXq
; this quantity is subadditive: Lpf ˝gq ď Lpf q `Lpgq. For p ě 1 we consider the moment conditions ż Lpf q p dνpf q ă `8, (M p ) Dp ą 1, pM p q holds, (M `) Dt ą 0,

ż }f } t C 1 pXq `› › f ´1› › t C 1 pXq dνpf q ă `8. (M exp )
When p " 1, (M p ) coïncides with the moment condition (M) from the introduction. For p ą 1, (M p ) implies (M `) which implies (M). The subadditivity of L and the convexity inequality

pr ´1 ř r i"1 L i q p ď r ´1 ř r i"1 L p i imply (2.2)
ż Lpf q p dν prq pf q ď r p ż Lpf q p dνpf q for p P r1, `8r and r P N ˚, where ν prq denotes the r th convolution power of ν.

Notation for random compositions.

Set Ω " Diff 1 pM q N ; its elements are sequences ω " pf n q ně0 of diffeomorphisms. We use the probabilistic notation Ep¨q and Pp¨q for the expectation and probability with respect to ν N on the probability space Ω. We let pF n q ně1 be the increasing sequence of σ-algebras in Ω generated by cylinders of length n, so that an event is F n -measurable if it depends only on the first n terms f 0 , . . . , f n´1 of ω " pf n q ně0 . For ω " pf n q ně0 P Ω we put f 0 ω " id and (2.3)

f n ω " f n´1 ˝¨¨¨˝f 0 for n ě 1; in particular f 1 ω " f 0 . For x in M and v P T x M zt0u we set (2.4)

x ω,n " f n ω pxq and v ω,n "

pf n ω q › pvq }pf n ω q › pvq} P T 1 xω,n M.
For any sequence of integers 0 " k 0 ă k 1 ă ¨¨¨ă k p " n the chain rule gives

(2.5) log }pf n ω q › v} " p´1 ÿ j"0 log › › › ´f k j`1 ´kj σ k j ω ¯› v ω,k j › › › 2.3.
Equivalent conditions for uniform expansion. Recall that the probability measure ν on Diff 1 pM q is uniformly expanding if there exists a real number c ą 0 and an integer n 0 ě 1 such that (2.6) for every v P T 1 X, ż log }f › pvq} dν pn 0 q pf q ě c.

Then, the cocycle relation for log }f›pvq} }v} implies that (2.7) ż log }f › pvq} dν pkn 0 q pf q ě kc for every k ě 1. Thus, ν is uniformly expanding if and only if ν pnq is uniformly expanding for some (and hence for all) n. Moreover, the uniform expansion property does not depend on our choice of a Riemannian metric on M .

Remark 2.1. If ν is uniformly expanding and the submanifold N Ă M is invariant under every diffeomorphism in the support of ν, then ν induces a uniformly expanding measure on Diff 1 pN q.

Lemma 2.2. Let ν be a probability measure on Γ satisfying (M). It is uniformly expanding if and only if

(2.8) @v P T 1 M, Dn " npvq such that ż log }f › v} dν pnq pf q ą 0.
This is Lemma 4.3.1 of [START_REF] Liu | Lyapunov exponents approximation, symplectic cocycle deformation and a large deviation theorem[END_REF], but Liu assumes that the support of ν is compact; thus we briefly reproduce his proof, assuming only (M).

Proof. We have to show that (2.8) implies (2.6). Since |log }f › v}| ď Lpf q for every v P T 1 X, the dominated convergence theorem implies that, for every n,

(2.9) v Þ Ñ ż log }f › pvq} dν pnq pf q
is continuous. Thus by compactness, there exists a finite open cover V 1 , . . . , V p of T 1 M , positive real numbers c i , and integers n i such that (2.10) ż log }f › pvq} dν pn i q pf q ě c i for every v P V i . Set c 0 " minpc i q and n 0 " maxpn i q. For v P T 1 X and ω P Ω, define the stopping time τ 1 pv, ωq to be the first integer n ě 1 such that ş log }f › v} dν pnq pf q ě c 0 , and then define inductively (2.11) τ k`1 pv, ωq " τ k pv, ωq `τ1 pv ω,k , σ k pωqq.

By construction, τ 1 does not depend on ω but τ k does; in addition τ k pv, ωq ď kn 0 for all k ě 1. For n ě 1, define K n pv, ωq, or Kpnq for short, by K n pv, ωq " max tk; τ k ď nu. Then Kpnq ě n{n 0 and n ´Kpnq ď n 0 ´1. With the convention τ 0 " 0, the chain rule (2.5) gives

E plog }pf n ω q › v}q " E ¨Kpnq´1 ÿ j"0 log › › › ´f τ j`1 ´τj σ τ j ω ¯› v ω,τ j › › › '`E ´´f n´Kpnq σ τ Kpnq ω ¯› v ω,τ Kpnq ě n n 0 c 0 ´max 1ďqďn 0 E pLpf q ω qq (2.12) ě n n 0 c 0 ´n0 ż Lpf qdνpf q. (2.13) Thus, for n ě n 0 2 `n2 0 c 0 ş
Lpf qdνpf q, we have E plog }pf n ω q › v}q ě c 0 2 ą 0 independently of v, as was to be shown. Lemma 2.3. Under the moment condition (M `), ν is uniformly expanding if and only if

(2.14) @v P T 1 X, Dc ą 0 such that P ˆ1 n log }pf n ω q › v} ě c ˙ÝÑ nÑ8 1.
Under the moment condition (M), Property (2.14) implies uniform expansion.

Proof. Let us first show that (2.14) implies (2.8) under the assumption (M). Fix v P T 1 X, set Ω n " ω P Ω ; 1 n log }pf n ω q › v} ě c ( , and split E `1 n log }pf n ω q › v} ˘into the sum of an integral over Ω n and an integral over Ω A n . The first one is larger than cPpΩ n q, and PpΩ n q tends to 1 as n goes to `8. The second one satisfies

ˇˇˇE ˆ1 n log }pf n ω q › v}1 Ω A n ˙ˇˇˇď E ˆ1 n Lpf n ω q1 Ω A n ˙. (2.15)
The moment condition and Kingman's subadditive ergodic theorem show that 1 n Lpf n ω q is uniformly integrable and converges almost surely to some finite constant; since PpΩ A n q converges to 0, we conclude that E `1 n log }pf n ω q › v} ˘ě c{2 for large n. For the converse implication we use a martingale convergence argument, as in [START_REF] Liu | Lyapunov exponents approximation, symplectic cocycle deformation and a large deviation theorem[END_REF]Lem. 4.3.5] and [22, Prop. 2.2](2 ). Choose p ą 1 such that (M p ) holds. For convenience, let us first replace ν by ν pn 0 q , where n 0 is given by the espansion property (2.6). Define (for some fixed unit vector v)

(2.16) X k " log › › pf 1 σ k ω q › v ω,k › › ´ż log }f › pv ω,k q}dνpf q.
These increments X k are uniformly bounded in L p because (2.17)

E `ˇl og › › pf 1 σ k ω q › v ω,k › › ˇˇp ˘1{p ď E `Lpf 1 σ k ω q p ˘1{p " ˆż Lpf q p dνpf q ˙1{p
and the second term in (2. [START_REF] Cantat | Quelques aspects des systèmes dynamiques polynomiaux: existence, exemples, rigidité[END_REF]) is pointwise bounded by (2.18) ˇˇˇż log }f › v ω,k }dνpf q ˇˇˇď ż Lpf qdνpf q ď ˆż Lpf q p dνpf q ˙1{p .

Thus, the sums S n " ř n´1 k"0 X k are all in L p . Since EpX n | F n q " 0 and S n is F n´1measurable, pS n q is a martingale relative to the filtration pF n´1 q. It follows from Theorem 2.22 in [35, §2.7] that 1 n S n converges to 0 in probability and in L p . Now, the chain rule gives

(2.19) 1 n S n pωq " 1 n log }pf n ω q › v} ´1 n ż log }f › v}dν pnq pf q,
and (2.7) asserts that ş log }f › v}dν pnq pf q ě cn, so we conclude that for any c 1 ă c

(2.20) P ˆ1 n log }pf n ω q › v} ě c 1 ˙ÝÑ nÑ8 1,
as desired. Recall however that we are working with ν pn 0 q : coming back to ν this means that (2.20) holds along the subsequence pnn 0 q. We then write n " kn 0 `r, with 0 ď r ď n 0 ´1, so that

(2.21) pf n ω q › v " pf r σ kn 0 ω q › pf kn 0 ω q › v
and what we have to show is that applying f r σ kn 0 ω does not affect the linear growth of log

› › pf kn 0 ω q › v › › . But the inequality (2.2), applied with p " 1, gives (2.22) P `D0 ď r ď n 0 ´1, ˇˇlog › › pf r σ kn 0 q › › › ˇˇě εk ˘ď n 0 ´1 ÿ r"0 ν prq pLpf q ě εkq ď Cn 2 0 εk ,
and we are done.

Remark 2.4. In the first part of the proof, the implication (2.14)ñ(2.8) is true for a given v, while the converse implication requires uniform expansion on the whole of X.

Remark 2.5. This proof shows that if ν satisfies (M 2 ), then the convergence in probability in (2.14) can be replaced by an almost sure convergence. (Indeed by Theorem 3 of [32, p. 243],

1 n S n converges almost surely to 0 when the X k are uniformly L 2 .)

INDUCING ON A FINITE INDEX SUBGROUP

3.1. Hitting times and hitting measures (see [START_REF] Benoist | Random walks on reductive groups[END_REF]Chap. 5]). Let ν be a probability measure on Diff 1 pM q and let G be the closed subsemigroup of Diff 1 pM q generated by ν. Let H Ă G be a closed finite index subsemigroup; this means that there is a continuous and transitive action G ˆF Ñ F on some finite set F such that H is the stabilizer of some element x 0 P F ; the index of H is rG : Hs " |F | and F is the quotient space.

The hitting time T H of H for the random walk induced by ν (starting from the neutral element) is

(3.1)
T H pωq " min tn ě 1, f n ω P Hu . Lemmas 5.4 and 5.5 in [START_REF] Benoist | Random walks on reductive groups[END_REF] show that T H is almost surely finite, admits an exponential moment, and satisfies EpT H q " rG : Hs. By definition the hitting measure (or induced measure) ν H is the probability measure on H describing the distribution of f . If H is a finite index semigroup and g P H, hg belongs to H if and only if h belongs to H. Thus, T H,k`1 pωq ´TH,k pωq " T H,1 pσ T H,k pωq pωqq and the Markov property implies that the random variables pT H,k`1 ´TH,k q are independent and identically distributed: each of them is distributed as T H . Since their expectation equals rG : Hs, the law of large numbers gives ( This result still holds if we substitute any subbaditive function to log }f } C 1 pXq in the definition of L (see Equation 2.1), with exactly the same proof.

Proof. Consider the finite quotient F of G by H and denote the action of G on F by left translations by (u Þ Ñ au, a P G); by definition H is the stabilizer of some x 0 P F . Set K " |F | " rG : Hs.

For each u P F , choose a sequence of measurable subsets A 1 puq, A 2 puq, . . ., A k puq in G, with k " kpuq ď K such that νpA i puqq ą 0 for each i and, for all sequences a i P A i puq, pa k ¨¨¨a 1 qu " x 0 while pa j ¨¨¨a 1 qu ‰ x 0 if j ă k. Since F is finite, there is a real number ε ą 0 such that νpA 1 puqq ¨¨¨νpA kpuq puqq ě ε for all u. Shrinking the A i puq if necessary, we may assume that Lpgq ď C for some C ą 0 and all g in Ť u,i A i puq. We split the integral of Lpf q p as a finite sum ş Lpf q p dνpf q " ř uPF ş tf x 0 "uu Lpf q p dνpf q. If pa kpuq , . . . , a 1 q P A kpuq puq ˆ¨¨¨ˆA 1 puq, then Lpf q ď Lpa kpuq ¨¨¨a 1 f q `KC because L is subadditive; thus, ż tf x 0 "uu Lpf q p dνpf q ď ż tf x 0 "uu pLpa kpuq ¨¨¨a 1 f q `KCq p dνpf q. (3.3) By construction, the product a kpuq ¨¨¨a 1 f is a first return in H. Thus, integrating over the A i puq, the distribution of a kpuq ¨¨¨a 1 f contributes positively to ν H , and we get

ε ż tf x 0 "uu Lpf q p dνpf q ď ż H pLpgq `KCq p dν H pgq. (3.4)
Assertion [START_REF] Arnold | Jordan normal form for linear cocycles[END_REF] follows from this estimate.

For assertion (2), we must bound ş Lpf q p 1 dν H pf q " EpLpf T H pωq ω q p 1 q. By subadditivity of L and convexity of s Þ Ñ s p 1 ,

E ˆL ´f T H pωq ω ¯p1 ˙ď E ¨¨T H pωq´1 ÿ i"0 Lpf i q 'p 1 ‹ ' (3.5) ď E ¨TH pωq p 1 ´1 T H pωq´1 ÿ i"0 Lpf i q p 1 '. (3.6)
Let r ą 1 and q ą 1 satisfy p " rp 1 and 1 q `1 r " 1. Set α " 1 q . We write

T p 1 ´1 H " T p 1 ´1`α H T ´α H
and apply Hölder's inequality to bound (3.6) from above by

ď E ´T pp 1 ´1`αqq H ¯1{q E ¨T ´αr H pωq ¨TH pωq´1 ÿ i"0 Lpf i q p 1 'r '1{r (3.7) ď E ´T pp 1 ´1`αqq H ¯1{q E ¨T ´αr H pωqT r{q H pωq T H pωq´1 ÿ i"0 Lpf i q p '1{r (3.8)
where in the last line we use the discrete Hölder inequality, rp 1 " p, and r ´1 " r{q. Since α " 1{q and r " p{p 1 the last expression reduces to (3.9)

E ˜T pp´1qp 1 p´p 1 H ¸1´p 1 {p E ¨TH pωq´1 ÿ i"0
Lpf i q p 'p 1 {p .

To conclude, we apply Lemma 5.4 of [START_REF] Benoist | Random walks on reductive groups[END_REF], which says that Ep ř T H pωq i"1 ϕ ˝σi q " EpT H qEpϕq for any integrable function ϕ, and we arrive at the bound (3.10)

ż Lpf q p 1 dν 1 pf q ď E ˜T pp´1qp 1 p´p 1 H ¸1´p 1 {p E pT H q p 1 {p ˆż Lpf q p dνpf q ˙p1 {p .
Since ν satisfies (M p ) and the hitting time T H admits moments of all orders, this last expression is finite, and the proof is complete. Assertion (3) follows from Assertions (1) and (2) and Corollary 5.6 in [START_REF] Benoist | Random walks on reductive groups[END_REF].

For the last assertion, fix an element h of H and an open neighborhood U of h in H. Since the action of G on F " G{H is continuous, there is an open neighborhood V of h in G such that every element of G X V is in U . Since the support of ν generates a dense subsemigroup of G the random walk induced by ν that starts at the neutral element visits V , hence the neighborhood U of h. Thus, ν H generates H.

3.2.

Uniform expansion of the induced measure. Proposition 3.2. Let ν be a probability measure on Diff 1 pM q satisfying (M). Assume that ν is uniformly expanding and let n 0 be as in (2.6). Then, the measure induced by ν pn 0 q on H is uniformly expanding.

In fact, the next proposition shows that, under condition (M `), ν is uniformly expanding if and only if ν H is. The proof of Proposition 3.2 is based on a simple martingale argument, while Proposition 3.3 relies on the criterion of Lemma 2.3.

Proof. We use ideas from [43, §4.3] and [START_REF] Chung | Stationary measures and orbit closures of uniformly expanding random dynamical systems on surfaces[END_REF]Prop. 2.2]. To ease notation we rename ν pn 0 q into ν so that (2.6) holds with n 0 " 1 and some c ą 0; as above, we denote by ν H the measure induced by ν (i.e. by ν pn 0 q ) on H. Fix v P T 1 X, and define a sequence of random variables pY k q kě0 by (3.11)

Y k pωq " log › › pf 1 σ k ω q › v ω,k › › ´c.
Then for all k ě 1, EpY k | F k q ě 0, so that the sequence pS n q ně1 defined by S n " ř n´1 k"0 Y k is a submartingale relative to the filtration pF n q: EpS n`1 | F n q ě S n . The moment condition (M) implies that Ep|S n`1 ´Sn | | F n q " Ep|Y n | | F n q is uniformly bounded. Since the hitting time T H is integrable, we can apply the optional stopping theorem [START_REF] Durrett | Probability: theory and examples[END_REF]Thm. 4.7.5], which implies that EpS T H q ě EpS 1 q ě 0. Unwinding the definitions and applying the chain rule, we see that (3.12) EpS T H q " ż log }f › v} dν H pf q ´crG : Hs, where we use EpT H q " rG : Hs. Therefore ş log }f › v} dν H pf q ě crG : Hs ą 0, and ν H is uniformly expanding. Proposition 3.3. Let ν be a probability measure on Diff 1 pM q satisfying (M `). Let ν H be the measure induced on a closed finite index subsemigroup. Then ν is uniformly expanding if and only if ν H is uniformly expanding.

Proof. Let us show that if ν H is uniformly expanding then ν is uniformly expanding. The converse implication is similar and is left to the reader (in this direction, Proposition 3.2 will actually be sufficient for our purposes). Fix v P T 1 M . In view of Lemma 2.3, we have to show that for some c ą 0,

(3.13) P ˆ1 n log }pf n ω q › v} ě c ˙ÝÑ nÑ8 1.
Consider the sequence of hitting times T H,k defined in § 3.1 and denote it by pT k q for simplicity (hence T 1 " T H ). By Theorem 3.1 ν H satisfies (M `), so we can apply Lemma 2.3 to get a real number c ą 0 such that

(3.14) P ˆ1 k log › › › ´f T k pωq ¯› v › › › ě c ˙ÝÑ kÑ8 1.
To obtain (3.13) we denote by Kpn, ωq :" max tk, T k pωq ď nu the number of visits of f j ω in H with 1 ď j ď n. If k ď Kpn, ωq, then We first conclude the proof of Proposition 3.3. Let Ω 1 n be as in Lemma 3.4. Let Ω 2 n be the set of itineraries ω such that the lower bound of (3.14) holds for k " t n γ ´n3{4 u. Then,

(3.15) pf n ω q › v " ´f n´T k pωq σ T k pωq ω ¯› ´f T k pωq ω ¯› v
PpΩ 1 n X Ω 2 n q Ñ 1 as n Ñ `8. Fix 0 ă ε ă γc and consider the set Ω 3 n Ă Ω 1 n X Ω 2 n made of all ω such that L ´f n´T k σ T k ω ¯ă εn for k " t n γ ´n3{4 u. Then (3.17) PppΩ 3 n q A q ď P ´L ´f n´T k σ T k ω ¯ě εn ¯ď P ˜max 0ďqďAn 3{4 q´1 ÿ i"0 L i ě εn
where pL i q iě0 is a sequence of independent random variables, each of them distributed as Lpgq for dνpgq. Since the L i are non-negative, PpΩ 3 n q ď P ´řAn 3{4 i"0 L i ě εn ¯. Now, the moment condition (M) and the Markov inequality give

(3.18) P ¨An 3{4 ÿ i"0 L i ě εn 'ď An 3{4 }Lpf q} L 1 pνq εn ď Cn ´1{4
for some C ą 0. Thus, PpΩ 1 n X Ω 2 n X Ω 3 n q Ñ 1 as n Ñ `8, and the conclusion follows since

1 n log }pf n ω q › v} ě γc ´ε for all ω P Ω 1 n X Ω 2 n X Ω 3 n .
Proof of Lemma 3.4. The following moderate deviations estimate follows from standard large deviations theory and the fact that T H has an exponential moment (see [23, 3.7

MARGULIS FUNCTIONS

In this section we develop some tools for the proof of the equidistribution Theorem 1.2. Under appropriate assumptions, we show that the measures ν n ˚δx and 1 n ř n k"1 δ f k ω pxq do not cluster at a Γ-periodic orbit, except when Γpxq is itself finite. The basic tool is the construction of a proper function, defined on the complement of such a periodic orbit, which essentially decreases along random trajectories. After [START_REF] Eskin | Recurrence properties of random walks on finite volume homogeneous manifolds[END_REF] it is often referred to as a "Margulis function", even if this strategy has a long history in the Markov chain literature (see [START_REF] Meyn | Markov chains and stochastic stability[END_REF]). Our presentation is greatly influenced by [START_REF] Benoist | Stationary measures and invariant subsets of homogeneous spaces (III)[END_REF] and [START_REF] Énard | Random walks with bounded first moment on finite-volume spaces[END_REF].

4.1.

A general recurrence criterion. For concreteness, instead of general Markov chains, we consider the setting of group actions. Theorem 4.1 (Bénard-De Saxcé [START_REF] Énard | Random walks with bounded first moment on finite-volume spaces[END_REF]). Let U be a locally compact topological space. Let Γ be a group of homeomorphisms of U , and ν be a probability measure on Γ. Assume that there exists a function u : U Ñ R `satisfying the assumptions: DA ą 0, Dγ ą 0, @x P U, upxq ě A ñ ż upf pxqqdνpf q ď upxq ´γ (4.1) DB ą 0, Dη ą 0, @x P U,

ż |upf pxqq ´upxq| 1`η dνpf q ď B. (4.2)
Then for every ε ą 0 there exists R ą 0 such that for all x in U , (1) there exists n x ě 0, such that pν n ˚δx qptu ě Ruq ď ε for all n ě n x ;

(2) for ν N -almost every ω,

lim sup nÑ8 1 n # ! k P t1, . . . , nu ; u ´f k ω pxq ¯ě R ) ď ε.
Furthermore the integer n x in (1) depends only on upxq.

For the proof, see Proposition 1.2 in [START_REF] Énard | Random walks with bounded first moment on finite-volume spaces[END_REF], and the comments following it. More precisely, we refer to [4, Prop. 2.5] for the conclusion (1), including the uniformity statement on n x , and to [4, Prop. 2.7] for [START_REF] Baouendi | Real submanifolds in complex space and their mappings[END_REF]. Even if this is not required in the proof, the function u has to be understood as a proper function on U , in which case the conclusions (1) and ( 2) correspond to a "non-escape of mass" property.

The original contraction property for the Margulis function u in [START_REF] Eskin | Recurrence properties of random walks on finite volume homogeneous manifolds[END_REF][START_REF] Benoist | Stationary measures and invariant subsets of homogeneous spaces (III)[END_REF] is Remark 4.2. For a holomorphic action on a compact complex manifold, these conditions are equivalent to their respective C 1 analogues (M p ) and (M `), because a uniform control on the first derivatives provides a uniform control of higher derivatives as well.

(4.3) D0 ă a ă 1, Db ą 0, @x P U, ż upf pxqqdνpf q ď aupxq `b instead of (4.
Theorem 4.3. Let Γ be a group of C 2 diffeomorphisms of a compact Riemannian manifold M , and ν be a measure on Γ satisfying the moment condition (M 2,`) . Let F be a finite orbit of Γ such that ν is uniformly expanding on F . Then for every x P M zF , for every ε ą 0 there exists a compact set K Ť M zF such that:

(1) pν n ˚δx qpKq ě 1 ´ε for n ě n x , and (2) for ν N -almost every ω,

lim sup nÑ8 1 n # ! k P t1, . . . , nu , f k ω pxq P K ) ě 1 ´ε.
Furthermore the integer n x in (1) is locally uniform in M zF .

This result seems to be new: it appears under stronger (exponential) moment assumptions in e.g. [START_REF] Liu | Lyapunov exponents approximation, symplectic cocycle deformation and a large deviation theorem[END_REF][START_REF] Chung | Stationary measures and orbit closures of uniformly expanding random dynamical systems on surfaces[END_REF]. Note that such a result is not expected to hold under the first moment condition (M 2,1 ), as explained in Examples 1 and 2 of Section 2 in [START_REF] Énard | Random walks with bounded first moment on finite-volume spaces[END_REF].

Proof. First, the proof of Proposition 3.3 in [START_REF] Énard | Random walks with bounded first moment on finite-volume spaces[END_REF] shows that if the conclusions (1) and ( 2) hold for ν pn 0 q , then they hold for ν. So we can replace ν by ν pn 0 q and hence assume that the uniform expansion property (1.1) holds (on F ) for n 0 " 1.

Let dp¨, ¨q be the Riemannian distance on M . According to Theorem 4.1, we only need to show that u :

x Þ Ñ ´log dpx, F q is a proper function M zF Ñ R `satisfying Properties (4.1) and (4.2). Preliminaries.-We set N pf q " }f } C 2 `› › f ´1› › C 2
and note that N pf q ě Lippf q `Lippf ´1q for every f P Γ. In particular, for every x P X

(4.5) 1 N pf q ď dpf pxq, F q dpx, F q ď N pf q.
For R ą 0, set ΓpRq " tf P Γ ; N pf q ď Ru. We choose η ą 0 such that the moment condition (M 2,p ) is satisfied with p " 1 `η. Then, (4.6)

I η :" ż Γ plogpN pf qqq 1`η dνpf q is a finite positive number. In what follows, we choose R ą 1 such that (4.7) 2I η plogpRqq η ă c 4
where c is the expansion factor in Equation (1.1) (along the finite orbit F ).

Take s ą 0 such that s is smaller than the injectivity radius of M at y, for every y P F ; -the balls Bpy; sq, for y in F , are pairwise disjoint; -C 0 R 2 s ă c{4, where c is the expansion factor as above, and C 0 is the constant appearing below in the Taylor expansion (Equation (4.9)).

Then, define V and V 1 by

(4.8) V " ď yPF Bpy; sq, V 1 " ď yPF Bpy; s{Rq.
By (4.5) we have f pV 1 q Ă V for every f P ΓpRq.

If x belongs to V , we denote by πpxq the unique point of F at distance ď s from x, and we denote by w x the unique vector in T πpxq M such that exp πpxq pw x q " x and }w x } " dpx, πpxqq. First estimate.-For f in ΓpRq and x P V 1 , Taylor's second order formula yields

(4.9) |dpf pxq, f pπpxqqq ´}f › pw x q}| ď C 0 N pf qdpx, πpxqq 2 ,
for some uniform constant C 0 , that does not depend on f . This gives

(4.10) ˇˇˇd pf pxq, F q dpx, F q ´}f › pw x q} }w x } ˇˇˇď C 0 N pf qdpx, F q.
Now, using the Lipschitz estimate (4.5) and the fact that |logpaq ´logpbq| ď N |a ´b| when a, b P rN ´1, N s, we obtain

(4.11) ˇˇˇl og ˆdpf pxq, F q dpx, F q ˙´log ˆ}f › pw x q} }w x } ˙ˇˇˇď C 0 N pf q 2 dpx, F q.
By the definition of ΓpRq and the requirements on s, we get

ż f PΓpRq ˇˇˇl og ˆdpf pxq, F q dpx, F q ˙´log ˆ}f › pw x q} }w x } ˙ˇˇˇd νpf q ď C 0 R 2 dpx, F q ď c 4 , (4.12)
because dpx, F q ď s. Second estimate.-Now, for any f in Γ we also have (4.13) ˇˇˇl og ˆdpf pxq, F q dpx, F q ˙´log ˆ}f › pw x q} }w x } ˙ˇˇˇď 2 logpN pf qq hence Markov's inequality and our choice of R give

ż f PΓpRq A ˇˇˇl og ˆdpf pxq, F q dpx, F q ˙´log ˆ}f › pw x q} }w x } ˙ˇˇˇd νpf q ď 2 logpRq η I η ď c 4 . (4.14)
Conclusion.-Summing the integrals over f in ΓpRq and ΓpRq A , we obtain (4.15)

ż f PΓ ˇˇˇl og ˆdpf pxq, F q dpx, F q ˙´log ˆ}f › pw x q} }w x } ˙ˇˇˇd νpf q ď c 2 .
Since w x is a vector tangent to M at πpxq P F , the uniform expansion along F yields (4.16)

ż log ˆ}f › pw x q} }w x } ˙dνpf q ě c
and then (4.15) implies that (4.17) ż ´log dpf pxq, F qdνpf q ď ´logpdpx, F qq ´c{2.

In other words, u : x Þ Ñ ´logpdpx, F qq satisfies Property (4.1) (with A " ´logpsq). Property (4.2) is obtained from (4.5) and the moment condition. Thus, as announced above, u satisfies the assumptions of Theorem 4.1, and we are done.

The local uniformity of n x in Theorem 4.3 has the following interesting consequence. Proof. Let µ be such a stationary measure. Fix ε ą 0, say ε " 1{2 and let K be as in Theorem 4.3. The stationarity of µ implies that for every n ě 0,

(
ν pn B q ptg, gx P Kuq ě 1 2 .
Plugging this into (4.19), we obtain 1 2 µpBq ď µpKq. Since B is arbitrary, this implies that µpM zF q ď 2µpKq and we are done.

4.3.

Totally real invariant manifolds. We now consider a situation which is specific to the complex setting. Theorem 4.5. Let X be a compact complex manifold of dimension d. Let Γ be a group of holomorphic diffeomorphisms of X, endowed with a probability measure ν satisfying (M `). Let Y Ă X be a Γ-invariant, analytic, totally real submanifold of maximal (real) dimension d, such that ν is uniformly expanding on Y . Then for any x P XzY and any ε ą 0, there exists a compact subset K Ť XzY such that the conclusions ( 1) and ( 2) of Theorem 4.3 hold.

The result also holds if Y admits finitely many singular points, under the stronger assumption that ν is finitely supported.

By "uniformly expanding along Y " we mean that the restriction of Γ to Y is uniformly expanding viewed as an action on Y , or equivalently that the uniform expansion condition (1.1) holds in X for every x P Y ; the equivalence between the two conditions comes from the fact that for every x P Y , the complex span of

T x Y is T x X. When Y is singular, we require that (1.1) holds in X along SingpY q.
Note also that this statement is specific to totally real submanifolds and holomorphic actions. In other words, there is no analogue of Theorem 4.1 when F is replaced by an arbitrary submanifold: see Example 4.6 below.

Proof of Theorem 4.5 when Y is smooth. We suppose Y smooth and show that there exists n ě 1 such that x Þ Ñ ´log dpx, Y q defines a Margulis function (i.e. satisfies (4.1) and (4.2)) for ν pnq . Then, as explained before, [START_REF] Énard | Random walks with bounded first moment on finite-volume spaces[END_REF] shows that (4.1) and (4.2) are automatically satisfied with n " 1. As in Theorem 4.1, Property (4.2) follows from the invariance of Y and the bilipschitz property; so we focus on (4.1).

For every x P Y there exists a local chart in which the equation of Y becomes Impzq " 0, where Impzq " Impz 1 , . . . , z d q " pImpz 1 q, . . . , Impz d qq (see [START_REF] Baouendi | Real submanifolds in complex space and their mappings[END_REF]Prop. 1.3.8 and 1.3.11]). We fix a finite family φ i : U i Ñ C d of such charts, covering a neighborhood of Y . The charts being bilipschitz, there exists an absolute constant D such that if x P U i , |log dpφ i pxq, φ i pY qq ´log dpx, Y q| ď D. Then from (2.7), replacing ν by ν pnq we may assume that the uniform expansion holds for n " 1 and the expansion constant c is bigger than 10D. This argument shows that it is enough to prove uniform expansion for ´log dp¨, Y q in the charts to infer the same property on X.

Let d U i denote the euclidean distance in the i-th chart (pulled back by φ i ). In U i , write φ i pxq " z " pz 1 , . . . , z d q and φ i pY q " tImpzq " 0u. Let πpφ i pxqq " pRepz 1 q, . . . , Repz d qq be the projection of φ i pxq on Y , so that (4.21) d U i px, Y q " }φ i pxq ´πpφ i pxqq} " }pImpz 1 q, . . . , Impz d qq} " }Impφ i pxqq}.

As before let ΓpRq " tf P Γ ; N pf q ď Ru, where

N pf q " }f } C 2 `› › f ´1› › C 2
, and fix f P ΓpRq. If x is sufficiently close to Y , then so does f pxq, hence f pxq belongs to some chart U j and working in this chart we get d U j pf pxq, Y q " }Impφ j pf pxqqq}. Applying Taylor's formula to the coordinate expression f of f , we obtain

φ j pf pxqq " f pφ i pxqq " f pπpφ i pxqqq `d fπpφ i pxqq pφ i pxq ´πpφ i pxqqq `O ´}φ i pxq ´πpφ i pxqq} 2 ¯.
Now, observe that the vector d fπpφ i pxqq pφ i pxq ´πpφ i pxqqq is purely imaginary because φ i pxq πpφ i pxqq is purely imaginary and d fπpφ i pxqq is real, since it preserves Y . Thus, taking imaginary parts and using (4.21) yields

(4.22) ˇˇˇd U j pf pxq, Y q d U i px, Y q ´› › df πpxq pv x q › › ˇˇˇď CRd U i px, Y q,
where v x " φ › i ´φi pxq´πpφ i pxqq }φ i pxq´πpφ i pxqq} ¯, πpxq " φ ´1 i πpφ i pxqq, and the constant C depends only on the charts. Arguing as in (4.11), plugging in the bilipschitz estimate for the distance to Y , and increasing C if necessary we get

(4.23) ˇˇˇl og dpf pxq, Y q dpx, Y q ´log › › df πpxq pv x q › › ˇˇˇď CR 2 dpx, Y q `2D.
Finally, using the moment condition to deal with the contribution of ΓzΓpRq as in Theorem 4.1, we obtain (4.24)

ż Γ ˇˇˇl og dpf pxq, Y q dpx, Y q ´log › › df πpxq pv x q › › ˇˇˇd νpf q ď CR 2 dpx, Y q `2D `C plog Rq η ,
and we conclude that log dp¨, Y q is a Margulis function by first fixing a large R and then choosing x sufficiently close to Y , as in Theorem 4.1.

Proof of Theorem 4.5 when Y is singular. When Y has finitely many singularities there is a priori no control of the distortion of the charts near SingpY q, so the argument must be modified. Fix x 0 P XzY and ε ą 0. We seek a compact subset K Ť XzY such that the conclusions (1) and ( 2) of Theorem 4.3 are satisfied for K and x 0 . Theorem 4.3 provides an open neighborhood V 1 of SingpY q such that these conclusions hold for

K 1 " V A 1 . Fix a neighborhood V 1 1 of SingpY q such that V 1 1 Ă V 1 ; let V be a small neighborhood of Y and set V 2 " V zV 1 1 .
We will construct a proper Margulis function u on XzpY X V 2 q. Then, Theorem 4.1 provides a compact set K 2 Ă XzpY X V 2 q such that (1) and (2) hold for K 2 . Therefore the desired conclusions hold for K :" K 1 X K 2 , with 2ε instead of ε.

To construct the desired Margulis function on XzpY X V 2 q, we put (4.25)

# upxq " ´log dpx, Y q for x P V 2 upxq " minpu| V 2 q ´B for x R V 2
where B " sup tlog N pf q, f P Supppνqu. Again, checking (4.2) is immediate, so we focus on (4.1). Since u is constant outside V 2 , we have to show that for x P V 2 sufficiently close to Y , (4.26)

ż Γ `´upf pxqq `upxq ´log › › df πpxq pv x › › ˘dνpf q ě ´c 2 ,
where c is the uniform expansion constant, and πpxq and v x are as above. As in the smooth case, the distortion between the distances in charts and the ambient distance is bounded on V 2 by a constant D " DpV 2 q; we iterate to get c ą 10D. We split (4.26) into (4.27)

ż Γ " ż tf PΓ, f pxqPV 2 u `żtfPΓ, f pxqRV 2 u
.

For the first integral we argue as in the smooth case to conclude that (4.28)

ż tf PΓ, f pxqPV 2 u ˇˇ´upf pxqq `upxq ´log › › df πpxq pv x › › ˇˇdνpf q ď c{2
when dpx, Y q is small enough. For the second integral we simply use the fact that if

f pxq R V 2 , then upxq ´upf pxqq ě B so ż tf PΓ, f pxqRV 2 u `´upf pxqq `upxq ´log › › df πpxq pv x › › ˘dνpf q (4.29) ě ż tf PΓ, f pxqRV 2 u `B ´log › › df πpxq pv x › › ˘dνpf q,
and this last term is non-negative from our choice of B. The proof is complete.

Example 4.6. There exists a group Γ " xf, gy of diffeomorphisms of the 3-torus R 3 {Z 3 and a finitely supported measure ν on Γ with xSupppνqy " Γ such that:

-Γ preserves Y :" R 2 {Z 2 ˆt0u; -there exists a neighborhood U of Y on which the dynamics of pΓ, νq is uniformly expanding; -for every x P U and almost every trajectory ω, f n ω pxq converges to Y .

Proof. Let 0 ă γ ă 1 and ψ be a diffeomorphism of the circle R{Z, fixing 0, and conjugate to t Þ Ñ γt on s ´1{4, 1{4rĂ R{Z by a diffeomorphism ϕ :s ´1{4, 1{4rÑ R such that ϕp0q " 0, ϕptq " t on r´1{8, 1{8s, and ϕpr´1{8, 1{8sq " r´1{4, 1{4s. Note that ψptq " γt on r´1{8, 1{8s and ψ preserves r´1{4, 1{4s (inducing a diffeomorphism of this interval). Pick A, B P SLp2, Zq generating a non-elementary subgroup, and define two diffeomorphisms g and h on R 3 {Z 3 by (4.30) gpx, y, zq " pApx, yq `pc 1 z, c 2 zq, ψpzqq and hpx, y, zq " pBpx, yq, ψpzqq.

We further assume that (4.31) pc 1 , c 2 q ‰ p0, 0q and γ is not an eigenvalue of A.

Let ν be a probability measure supported on g, h, g ´1, h

´1( such that 0 ă νpg ´1q ă νpgq and 0 ă νph ´1q ă νphq. Then there exists Ω 0 Ă Ω of full ν N -measure such that for every p " px, y, zq P R 2 {Z 2 ˆs ´1{4, 1{4r, and ω P Ω 0 , f n ω ppq Ñ Y . Indeed, writing f n ω ppq " px n , y n , z n q, we have:

(1) z n Ps ´1{4, 1{4r because ψ preserves s ´1{4, 1{4r;

(2) ϕpz n q " γ ř n i"1 ε i ϕpzq, where pε n q is a sequence of independent random variables with Ppε " 1q " νpgq`νphq and Ppε " ´1q " νpg ´1q`νph ´1q. Since, νpg ´1q`νph ´1q ă νpgq `νphq, φpz n q converges almost surely to 0.

To conclude, we have to show that the dynamics of pΓ, νq is uniformly expanding in R 2 {Z 2 ˆs1 {4, 1{4r. Indeed, if p P R 2 {Z 2 ˆs ´1{4, 1{4r and ω P Ω 0 , there is npωq such that f n ω ppq P R 2 {Z 2 ˆs ´1{8, 1{8r for n ě npωq. Now, in R 2 {Z 2 ˆs ´1{8, 1{8r the dynamics is linear, and the tangent action is generated by (4.32) g "

¨A ˆc1 c 2 0 γ ' and h " ˆB 0 0 γ ˙.
We claim that the linear action of p Γ, νq on R 3 is uniformly expanding, where Γ " xg, hy and ν is the measure naturally corresponding to ν. Indeed, the action is uniformly expanding on R 2 ˆt0u and if it were not uniformly expanding on R 3 , by Furstenberg-Kifer [START_REF] Furstenberg | Random matrix products and measures on projective spaces[END_REF], there would exist a Γ-invariant line transverse to R 2 ˆt0u along which the Lyapunov exponent would be non-positive. But the hypotheses (4.31) guarantee that such a line does not exist. From this, we deduce that there exists c ą 0 (any constant smaller than the Lyapunov exponent of the random product generated by A and B will do) such that for every p P R 2 {Z 2 ˆs ´1{4, 1{4r, every unit tangent vector v at p and almost every ω, 1 n log }pf n ω q › v} ě c if n is large enough. Applying Lemma 2.3 finishes the proof.

AN ERGODIC-THEORETIC CRITERION FOR EXPANSION

5.1. Construction of stationary measures. Let M be a compact manifold endowed with a riemannian metric; let T 1 M denote its unit tangent bundle and π : T 1 M Ñ M be the canonical projection. As in Section 2, if f is a diffeomorphism of M , we denote by f › its action on T M . Let ν be a probability measure on Diff 1 pM q satisfying the moment condition (M). We apply a classical strategy to get the following theorem (see e.g. [START_REF] Chung | Stationary measures and orbit closures of uniformly expanding random dynamical systems on surfaces[END_REF]Prop. 3.17], and [START_REF] Hurtado | The Burnside problem for DiffωpS 2 q[END_REF]Lem. 3.3]).

Theorem 5.1. Assume that there exists an increasing sequence pn k q P N N and a sequence of unit tangent vectors pu k q P pT 1 M q N such that

(5.1) lim kÑ8 1 n k ż log }f › u k }dν pn k q pf q " χ 0 .
Then, there exists a real number χ ě χ 0 , an ergodic ν-stationary probability measure μ on T 1 M , and a ν-almost surely invariant sub-bundle V Ă T M such that the top Lyapunov exponent of the projected measure µ :" π › μ in restriction to V is equal to χ. Likewise, there exists a real number χ 1 ď χ 0 that satisfies the same property for some pair pμ 1 , V 1 q.

Note that if μ is a probability measure on T 1 M that is ν-stationary for the tangent action, then its projection µ on M is ν-stationary as well; and if μ is ergodic, so is µ. When χ ą 0, one typically obtains V " T M .

Proof (see [START_REF] Chung | Stationary measures and orbit closures of uniformly expanding random dynamical systems on surfaces[END_REF][START_REF] Hurtado | The Burnside problem for DiffωpS 2 q[END_REF]). Consider the sequence of measures μk on T 1 M defined by (5.2) μk "

1 n k n k ´1 ÿ j"0 ν pjq ‹ δ u k " 1 n k n k ´1 ÿ j"0 ż f › u k }f › u k } dν pjq pf q,
where ν pjq ‹ δ u k denotes the convolution for the action of DiffpM q on the unit tangent bundle.

Since T 1 M is compact and the μk are probability measures, we can extract a subsequence (still denoted by μk for simplicity) that converges weakly towards a probability measure μ8 on T 1 M . By construction, this measure is ν-stationary.

The function Dilpf, uq :" log }f › u} is continuous on Diff 1 pM q ˆT 1 M . For u P T 1 M the chain rule gives

1 n ż log }pf n ω q › u}dν N pωq " 1 n n´1 ÿ j"0 ż Dil ¨fj , pf j ω q › u › › ›pf j ω q › u › › › 'dν N pωq " ż gPDiffpM q ˜1 n n´1 ÿ j"0 ż Dil ˆg, h › u }h › u} ˙dν pjq phq ¸dνpgq (5.3)
If we apply this equation to n " n k and u " u k the term between parentheses in the last integral is equal to ş Dil pg, uq dμ k puq, so, letting k go to `8, we conclude that

(5.4) lim kÑ8 1 n k ż log }f › u k }dν pn k q pf q " χ 0 " ż Diff 1 pM q ż T 1 M Dilpg, uqdμ 8 puqdνpgq
Thus, there exists χ ě χ 0 (resp. χ ď χ 0 ) and an ergodic component μ of μ8 such that (5.5)

ż Diff 1 pM q ż T 1 M
Dilpg, uqdμpuqdνpgq " χ.

As observed above, µ " π › μ is an ergodic ν-stationary probability measure. Denote by μx the conditional measures obtained by disintegration of μ with respect to the fibers of π, that is, μ " ş μx dµpxq. For µ-almost every x, let V pxq be the linear span of Supppμpxqq. Since Supppμq is ν-almost invariant and f › acts linearly along the fibers of T M , we infer that V is a νalmost invariant measurable sub-bundle. The Furstenberg formula asserts that the top Lyapunov exponent of µ in restriction to V is equal to χ. For completeness let us recall the argument: the ergodic theorem shows that for pν N ˆµq-almost every pω, xq and μx -almost every u P T 1

x M , (5.6)

lim nÑ`8 1 n n´1 ÿ j"0 Dil `fj , pf j ω q › u ˘" ż Diff 1 pM q N ż T 1 M Dilpf 1 ω , uq dμpuqdν N pωq " χ
where as usual ω " pf 0 , f 1 , . . .q, f 1 ω " f 0 , and f j ω " f j´1 ˝¨¨¨˝f 0 . On the other hand the Oseledets theorem asserts that for pν N ˆµq-almost every pω, xq, there exists a proper subspace W pω, xq Ă V pxq such that for u R W pω, xq, 1 n log }pf n ω q › u} converges to the top Lyapunov exponent χ `pµ, V q of µ in restriction to V . Thus by (5.6), χ `pµ, V q " χ, and the proof is complete. 5.2. Application I: positive exponent from cohomology. Fix a norm }¨} on the cohomology H ˚pM ; Rq " À m k"0 H k pM ; Rq, and denote also by }¨} the associated operator norm. As usual we denote by f › the action of a diffeomorphism f on differential forms as well as on H ˚pM ; Rq. For every 0 ď q ď m, the probability measure ν induces a random product of matrices on H q pM ; Rq, whose top Lyapunov exponent is by definition (see [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF]) (5.7) λ H q pM,Rq " λ H q pM,Rq pνq :" lim nÑ`8

1 n ż log }f › }dν pnq pf q.

We define λ H ˚pM,Rq " max 0ďqďm λ H q pM,Rq and refer to this exponent as the top cohomological exponent of ν. By the moment condition (M) this exponent is finite (see [19, §5.3]), and since DiffpM q preserves the image of the integral cohomology H ˚pM ; Zq, it is also nonnegative.

Lemma 5.2. For every 1 ď q ď m, there exists a sequence of unit tangent vectors pu n q P pT 1 M q N such that

lim inf nÑ8 1 n ż log }f › u n }dν pnq pf q ě 1 q λ H q pM,Rq .
This follows directly from the definitions, once one fixes a family of closed q-forms pα j q 1ďjďbqpM q , whose cohomology classes form a basis of H q pM ; Rq. Hence Theorem 5.1 entails: Corollary 5.3. If the top cohomological exponent of ν is positive there exists an ergodic stationary measure µ on M with a positive Lyapunov exponent.

Thus, for automorphisms of Kähler surfaces, Proposition 5.2 of [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF] gives: Corollary 5.4. Let X be a compact Kähler surface. Let ν be a probability measure on AutpXq satisfying the moment condition (M). If Γ ν is non-elementary there exists a ν-stationary measure µ on X with a positive Lyapunov exponent.

Application II:

Chung's criterion. The following theorem, taken from [22, Prop 3.17], plays an important role in this paper; a variant of this result appears in [START_REF] Brown | Zimmer's conjecture: Subexponential growth, measure rigidity, and strong property (T)[END_REF]. It is stated in [START_REF] Chung | Stationary measures and orbit closures of uniformly expanding random dynamical systems on surfaces[END_REF] for C 2 actions on surfaces but it holds in greater generality. The proof follows directly from the second assertion of Theorem 5.1.

Theorem 5.5 (Chung). Let M be a compact manifold. Let ν be a probability measure on Diff 1 pM q that satisfies (M). If ν is not uniformly expanding there exists an ergodic ν-stationary measure µ on M and a µ-measurable subbundle W Ă T M such that

(a) 0 ă dimpW q ď dimpM q; (b) W is ν-almost surely invariant; (c) in restriction to W , the top Lyapunov exponent of µ is non-positive.
Conversely, if such a pair pµ, W q exists, then ν is not uniformly expanding.

When M is a surface and ν is supported by the group of diffeomorphisms preserving some fixed area form the Lyapunov exponents of any ergodic stationary measure µ satisfy λ `pµq λ´p µq " 0. Thus, in Chung's theorem, either λ ´pµq " λ `pµq " 0 and we can take W " T M or λ ´pµq ă 0 ă λ `pµq and W coïncides with the stable line field provided by the Oseledets theorem; thus, µ is not hyperbolic or it is hyperbolic and its stable line field is non-random.

Part 2. Non-elementary actions on complex surfaces

From now on we denote by X a compact complex surface, endowed with a group Γ of holomorphic diffeomorphisms. Recall from [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF] that if Γ is non-elementary, then X is necessarily projective and Γ Ă AutpXq.

PRELIMINARIES

In this section we briefly recall some results from [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF] (see also [START_REF] Cantat | Sur la dynamique du groupe d'automorphismes des surfaces K3[END_REF][START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF]).

6.1. Parabolic automorphisms and their dynamics (see [20, §3]). Let h be a parabolic automorphism of a compact projective surface X (most of this discussion is valid for a compact Kähler surface). Then, h preserves a genus 1 fibration π h : X Ñ B, and every h-invariant holomorphic (singular) foliation -in particular any invariant fibration -coïncides with π. Let h B denote the automorphism of B such that (6.1)

π ˝h " h B ˝π.
If X is not a torus there is a positive integer m such that h m preserves every fiber of π, i.e. h m B " id B . When h B " id B we say that h is a Halphen twist. The set of Halphen twists in a given subgroup Γ Ă AutpXq is denoted by HalpΓq. Remark 6.1. If Γ is non-elementary and contains a Halphen twist (resp. a parabolic automorphism) h, then the conjugacy class of h in Γ contains Halphen twists (resp. parabolic automorphisms) associated with infinitely many distinct invariant fibrations, as follows from [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]Lem 2.17].

Suppose now that h is a Halphen twist. Then, h acts by translation on every smooth fiber of π (see [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]Prop. 2.18]). To be more precise, denote by Critpπq Ă B the finite set of critical values of π and set B ˝" Bz Critpπq. Fix some simply connected open subset U Ă B ˝, endowed with a section σ of π and a continuous choice of basis for H 1 pX w , Zq. Each fiber X w :" π ´1pwq, w P U , is an elliptic curve with zero σpwq, and one can find a holomorphic function τ on U , with values in the upper half plane, such that X w is isomorphic to C{Latpwq for Latpwq " Z ' Zτ pwq. On X w , h is a translation h w pzq " z `tpwq, for some holomorphic function w P U Þ Ñ tpwq P C{Latpwq. Moreover, Lemma 6.2(4) says that h behaves like a "complex Dehn twist", with a shearing property in the direction which is transversal to the fibers; thus shearing (or twisting) occurs along X w whenever t and τ are "transverse" at w (see § 9.1 for more details on the non-twisting locus).

The points w for which h w is periodic are characterized by the relation tpwq P Q ' Qτ pwq.

If (6.2)
tpwq ´pα `βτ pwqq P R ¨pp `qτ pwqq for some pα, βq P Q 2 and pp, qq P Z 2 , the closure of Ztpwq in C{Latpwq is an abelian Lie group of dimension 1, isomorphic to Z{kZ ˆR{Z for some k ą 0; then, the closure of each orbit of h w is a union of k circles. This occurs along a countable union of analytic curves R α,β p,q Ă U .

Otherwise, the orbits of h w are dense in X w , and the unique h w invariant probability measure is the Haar measure on X w .

The following lemma summarizes this discussion. Lemma 6.2. Let h be a Halphen twist with invariant fibration π : X Ñ B. Then,

(1) h acts by translation on each fiber X w " π ´1pwq, w P B ˝;

(2) for w in a dense countable subset of B ˝, the orbits of h w are finite;

(3) there is a dense, countable union of analytic curves R j in B ˝, such that: (a) for w R Ť j R j , the action of h in the fiber X w is a totally irrational translation (it is uniquely ergodic, and its orbits are dense in X w ); (b) for w P Ť j R j the orbits of h w are either finite or dense in a finite union of circles; (4) there is a finite subset NT h such that for x R π ´1 pNT h q

lim nÑ˘8 }D x h n } Ñ `8
locally uniformly in x; more precisely for every v P T x XzT x X πpxq , }D x h n pvq} grows linearly while 1 n π › pD x h n pvqq converges to 0. If moreover h preserves a totally real 2-dimensional real analytic subset Y Ă X, then:

(5) the generic fibers of π| Y are union of circles, there exists an integer m such that h m preserves each of these circles, and h m is uniquely ergodic along each of these circles, except for countably many fibers.

Property (4) is the above mentioned twisting property of h. Property (5) occurs, for instance, when X and h are defined over R and Y " XpRq is the real part of X. There are also examples of subgroups Γ Ă AutpXq preserving a totally real surface Y Ă X which is not the real part of X for any real structure, see [20, §9]). 6.2. Classification of invariant measures. Recall from Example 1.4 that if X is a torus, a K3 surface, or an Enriques surface it admits a canonical AutpXq-invariant volume form vol X . The associated probability measure will also be denoted by vol X . Such an area form exists also on any totally real surface, by virtue of the following lemma. Lemma 6.3 (see [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF]Remark 2.3]). Let X be an Abelian surface, or a K3 surface, or an Enriques surface with universal cover X. Let Y Ă X be a totally real surface of class C 1 , and AutpX; Y q be the subgroup of AutpXq preserving Y . If Y is totally real, the canonical holomorphic 2-form Ω X (resp. Ω X ) induces a smooth AutpX; Y q-invariant probability measure vol Y on Y . Theorem 6.4 (see [20, Thm A]). Let X be a projective surface. Let Γ be a non-elementary subgroup of AutpXq containing a parabolic element. Let µ be a Γ-invariant ergodic probability measure on X. Then, µ satisfies exactly one of the following properties.

(a) µ is the average on a finite orbit of Γ; (b) µ is non-atomic and supported on a Γ-invariant algebraic curve D Ă X;

(c) there is a Γ-invariant proper algebraic subset Z of X, and a Γ-invariant, totally real analytic surface Y of XzZ such that (1) µpY q " 1 and µpZq " 0; (2) Y has finitely many irreducible components; (3) the singular locus of Y is locally finite in XzZ; (4) µ is absolutely continuous with respect to the Lebesgue measure on Y ; and (5) its density (with respect to any real analytic area form on the regular part of Y ) is real analytic; (d) there is a Γ-invariant proper algebraic subset Z of X such that (1) µpZq " 0, (2) the support of µ is equal to X; (3) µ is absolutely continuous with respect to the Lebesgue measure on X; and (4) the density of µ with respect to any real analytic volume form on X is real analytic on XzZ.

If X is not a rational surface, then in case (c) (resp. (d)) we can further conclude that the invariant measure is proportional to vol Y (resp. vol X ).

HYPERBOLICITY OF INVARIANT MEASURES

Here, X is a compact Kähler surface. We fix a Kähler form κ 0 on X; norms of tangent vectors and differentials will be computed with respect to it. 7.1. Ledrappier's invariance principle and invariant measures on PT X . In this paragraph we collect some preliminary results for the proof of Theorems 1.3 and 7.4. Our presentation is inspired by [START_REF] Barrientos | Extremal exponents of random products of conservative diffeomorphisms[END_REF]. It is similar in spirit to that of [START_REF] Obata | Positive exponents for random products of conservative surface diffeomorphisms and some skew products[END_REF], which relies on the "pinching and twisting" formalism of Avila and Viana (see [START_REF] Viana | Lectures on Lyapunov exponents[END_REF] for an introduction 3 ). Most of this discussion is valid for a random holomorphic dynamical system on an arbitrary complex surface (not necessarily compact), satisfying (M).

We denote by PT X the projectivized tangent bundle of X; if f is an automorphism of X, we denote by PpDf q the induced action on PT X.

Let ν be a probability measure on AutpXq that satisfies the moment condition (M). We endow Ω :" AutpXq N (resp. Σ :" AutpXq Z ) with the probability measure ν N (resp. ν Z ), and set X `" Ω ˆX (resp. X " Ω ˆX); σ will denote the shift (on Ω or Σ). For ω " pf i q iě0 P Ω, we keep the notation f n ω from § 2.2. Then, we define F `: X `Ñ X `by F `pω, xq " pσpωq, f 1 ω pxqq; F : X Ñ X is defined by the same formula. For further standard notations, we refer to [19, §7].

Let µ be an ergodic ν-stationary measure on X. We introduce the projectivized tangent bundles PT X `" Ω ˆPT X and PT X " Σ ˆPT X. The bundles T X and PT X admit measurable trivializations over a set of full measure. Consider any probability measure μ on PT X that is stationary under the random dynamical system induced by pX, νq on PT X and whose projection on X coincides with µ, i.e. π › μ " µ where π : PT X Ñ X is the natural projection.

Such measures always exist: indeed the set of probability measures on PT X projecting to µ is compact and convex, and it is non-empty since it contains the measures ş δ rvpxqs dµpxq for any measurable section x Þ Ñ rvpxqs of PT X. Thus, the operator ş PpDf q dνpf q has a fixed point on that set. The stationarity of μ is equivalent to the invariance of ν N ˆμ under the transformation p F `: Ω ˆPT X Ñ Ω ˆPT X defined by (7.1) p F `pω, x, rvsq " pσpωq, f 1 ω pxq, PpD x f 1 ω qrvsq for any non-zero tangent vector v P T x X. We denote by μx the family of probability measures on the fibers PT x X of π given by the disintegration of μ with respect to π. The conditional measures of ν N ˆμ with respect to the projection PT X `Ñ X are given by μω,x " ν N ˆμ x .

Remark 7.1. Even when µ is Γ ν -invariant, this construction only provides a stationary measure on PT X. This is exactly what happens for non-elementary subgroups with a parabolic automorphism: indeed, we will show in § 7.2 that projectively invariant measures do not exist in this case.

The tangent action of our random dynamical system gives rise to a stationary product of matrices in GLp2, Cq. To see this, fix a measurable trivialization P : T X Ñ X ˆC2 , given by linear isomorphisms P x : T x X Ñ C 2 . It conjugates the action of DF `to that of a linear cocycle A : X `ˆC 2 Ñ X `ˆC 2 over pX `, F `, ν N ˆµq. In this context, Ledrappier establishes in [START_REF] Ledrappier | Positivity of the exponent for stationary sequences of matrices[END_REF] the following "invariance principle". Theorem 7.2. If λ ´pµq " λ `pµq, then for any stationary measure μ on PT X projecting to µ, we have PpD x f q › μx " μfpxq for µ-almost every x and ν-almost every f . The second ingredient in the proof of Theorem 1.3 is a description of such projectively invariant measures; this is where we follow [START_REF] Barrientos | Extremal exponents of random products of conservative diffeomorphisms[END_REF]. To explain this result a bit of notation is required. Let V and W be hermitian vector spaces of dimension 2; we fix two isometric isomorphisms ι V : V Ñ C 2 and ι W : W Ñ C 2 to the standard hermitian space C 2 , and we endow the projective lines PpV q and PpW q with their respective Fubini-Study metrics. If g : V Ñ W is a linear isomorphism, we set (7.2)

g " }Ppgq} C 1

where Ppgq : PpV q Ñ PpW q is the projective linear map induced by g and }¨} C 1 is the maximum of the norms of D z Ppgq : T z PpV q Ñ T Ppgqpzq PpW q with respect to the Fubini-Study metrics.

If ι W ˝g ˝ι´1 V " k 1 ak 2 is the KAK decomposition of ι W ˝g ˝ι´1 V in PSLp2, Cq, we get g " }a} 2 " › › ι W ˝g ˝ι´1 V › › 2
where }¨} is the matrix norm in PSL 2 pCq " SL 2 pCq{x˘idy associated to the Hermitian norm of C 2 . In particular, (a) g " 1 if and only if Ppgq is an isometry from PpV q to PpW q; (b) for a sequence pg n q of linear maps V Ñ W , g n tends to `8 with n if and only if Ppι W ˝g ˝ι´1 V q diverges to infinity in PSL 2 pCq. If f is an automorphism of X and x is a point of X, then κ 0 endows T x X and T f pxq X with hermitian structures, and we can apply this discussion to D x f : T x X Ñ T f pxq X. We are now ready to state the classification of projectively invariant measures.

Theorem 7.3. Let pX, νq be a random dynamical system on a complex surface and let µ be an ergodic stationary measure. Let μ be a stationary measure on PT X such that π › μ " µ and PpD x f q › μx " μfpxq for µ-almost every x and ν-almost every f . Then, exactly one of the following two properties is satisfied:

(1) For pν N ˆµq-almost every pω, xq, the sequence D x f n ω is unbounded and then: (1.a) either there exists a measurable Γ ν -invariant family of lines Epxq Ă T x X such that μx " δ rEpxqs for µ-almost every x; (1.b) or there exists a measurable Γ ν -invariant family of pairs of lines E 1 pxq, E 2 pxq Ă T x X and positive numbers λ 1 , λ 2 with λ 1 `λ2 " 1 such that μx " λ 1 δ rE 1 pxqs `λ2 δ rE 2 pxqs for µ-almost every x.

(2) The projectivized tangent action of Γ ν is reducible to a compact group, that is there exists a measurable trivialization of the tangent bundle pP x : T x X Ñ C 2 q xPX , such that for almostevery f P Γ ν and every x, P `Pfpxq ˝Dx f ˝P ´1

x ˘belongs to the unitary group PU 2 pCq.

In assertion (1.b), the pair is not ordered: there is no natural distinction of E 1 and E 2 , the elements of Γ ν may a priori permute these lines. The proof is obtained by adapting the arguments of [START_REF] Barrientos | Extremal exponents of random products of conservative diffeomorphisms[END_REF] to the complex case; the details are given in § 7.4.

7.2.

Proof of Theorem 1.3. By Theorem 6.4, µ is either equivalent to the Lebesgue measure on X, or to the 2-dimensional Lebesgue measure on some components of an invariant totally real surface Y Ă X.

7.2.1.

Proof of the hyperbolicity of µ. Let us assume, by way of contradiction, that µ is not hyperbolic. Hence its Lyapunov exponents vanish, and by Theorem 7.2 and Theorem 7.3, there is a measurable set X 1 Ă X with µpX 1 q " 1 such that one of the following properties is satisfied along X 1 : (a) there is a measurable Γ ν -invariant line field Epxq; (b) there exists a measurable Γ ν -invariant splitting Epxq ' E 1 pxq " T x X of the tangent bundle; here, the invariance should be taken in the following weak sense: an element f of Γ ν maps Epxq to Epf pxqq or E 1 pf pxqq; (c) there exists a measurable trivialization P x : T x X Ñ C 2 such that in the corresponding coordinates the projectivized differential PpDf x q takes its values in PU 2 pCq for all f P Γ ν and µ-almost all x P X 1 .

Fix a small ε ą 0. By Lusin's theorem, there is a compact set K ε with µpK ε q ą 1 ´ε such that the data x Þ Ñ Epxq, or x Þ Ñ pEpxq, E 1 pxqq or x Þ Ñ P x in the respective cases (a), (b), and (c) are continuous on K ε . In particular, in case (c), the norms of P x and P ´1

x are bounded by some uniform constant Cpεq on K ε ; hence, if g P Γ ν and x and gpxq belong to K ε , Dg x is bounded by Cpεq 2 .

Fix a pair of parabolic elements g and h P Γ ν with distinct invariant fibrations π g : X Ñ B g and π h : X Ñ B h respectively (see Remark 6.1). These two fibrations are tangent along some curve Tangpπ g , π h q in X.

' In a first stage we assume that X is not a torus. According to Section 6.1, there is an integer N ą 0 such that g N and h N preserve every fiber of their respective invariant fibrations. From now on, we replace g by g N and h by h N .

First assume that µ is absolutely continuous with respect to the Lebesgue measure on X, with a positive real analytic density on the complement of some invariant, proper, Zariski closed subset. We apply Lemma 6.2 to h and remark that pπ h q › µ can not charge the union of the curves R j . Then, we disintegrate µ with respect to π h to obtain conditional measures µ b , for b P B h ; since π h is holomorphic, the measures µ b are absolutely continuous with respect to the Haar measure on almost every fiber π ´1 h pbq. By Lemma 6.2, there exists a fiber π ´1 h pbq such that (1) the Haar measure of K ε X π ´1 h pbq is positive, (2) b R NT h and (3) the dynamics of h in π ´1 h pbq is uniquely ergodic. These properties hold for b " π h pzq, for µ-almost all z in K ε . Then we can pick x P π ´1 h pbq such that ph k pxqq kě0 visits K ε infinitely many times (4 ). The fifth assertion of Lemma 6.2 rules out case (c) because the twisting property implies that the projectivized derivative Dh n

x tends to infinity, while it should be bounded by Cpεq 2 when h n pxq P K ε . Case (b) is also excluded: under the action of h n , tangent vectors projectively converge to the tangent space of the fibers, so the only possible invariant subspace of dimension 1 is kerpDπ h q. Thus we are in case (a) and moreover Epxq " ker D x π h for µ-almost every x. But then, using g instead of h and the fact that µ does not charge the curve Tangpπ g , π h q, we get a contradiction. This shows that the last alternative (a) does not hold either, and this contradiction proves that µ is hyperbolic.

If µ is supported by a 2-dimensional real analytic subset Y Ă X, the same proof applies, except that we disintegrate µ along the singular foliation of Y by circles induced by π h and we use the fact that a generic leaf is a circle along which h is uniquely ergodic (see Lemma 6.2.( 4)).

' If X is a torus its tangent bundle is trivial and the differential of an automorphism is constant. In an appropriate basis, the differential of a Halphen twist h is of the form

(7.3) ˆ1 α 0 1 ˙with α ‰ 0.
Thus we are in case (a) with Epxq " ker D x π h for µ-almost every x. Using another twist g transverse to h we get a contradiction as before.

7.2.2. Proof of the positivity of the fiber entropy. This follows from classical arguments. Since µ is invariant the measure m " ν Z ˆµ on X is F -invariant. In both cases µ ! vol X and µ ! vol Y , respectively. the absolute continuity of the foliation by local Pesin unstable manifolds implies that the unstable conditionals of m cannot be atomic, see e.g. [START_REF] Ledrappier | Entropy formula for random transformations[END_REF]. Since the unstable conditionals of a zero entropy stationary measure are automatically atomic (see [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]Cor. 7.15]), we conclude that µ has positive fiber entropy. This concludes the proof of Theorem 1. (see [18, §4] for a thorough treatment)). Let A " C 2 {Λ be a complex torus and let η be the involution given by ηpz 1 , z 2 q " p´z 1 , ´z2 q; it has 16 fixed points. Then A{xηy is a surface with 16 singular points, and resolving these singularities (each of them requires a single blow-up) yields a Kummer surface X. Let f A be a loxodromic automorphism of A which is induced by a linear transformation of C 2 preserving Λ; then f A commutes to η and goes down to an automorphism f of X; such automorphisms will be referred to as loxodromic, classical, Kummer examples. They preserve the canonical volume vol X . The Kummer surface X also supports automorphisms which do not come from automorphisms of A (see [START_REF] Keum | The automorphism groups of Kummer surfaces associated with the product of two elliptic curves[END_REF] and [START_REF] Dolgachev | Birational automorphisms of quartic Hessian surfaces[END_REF] for instance).

In the following statement we do not assume that Γ ν contains a parabolic element (the approach is similar to that of [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]Thm 11.4]).

Theorem 7.4. Let pX, νq be a non-elementary random dynamical system on a Kummer K3 surface satisfying (M) and such that Γ ν contains a loxodromic classical Kummer example. Then any ergodic Γ ν -invariant measure giving no mass to proper Zariski closed subsets of X is hyperbolic.

Proof. The proof is similar to that of Theorem 1.3 so we only sketch it. Assume by contradiction that µ is not hyperbolic; since X is a K3 surface, the volume invariance shows that the sum of the Lyapunov exponents of µ vanishes (see [19, §7.3]); thus, each of them is equal to 0, and one of the alternatives of Theorem 7.3 holds, referred to as (a), (b), (c) as in Section 7.2, page 29.

By assumption, Γ ν contains a loxodromic, classical Kummer example f associated to a linear automorphism f A of a torus A. This automorphism f is uniformly hyperbolic in some dense Zariski open subset U , which is thus of full µ-measure: its complement is given by the sixteen rational curves coming from the resolution of the singularities of A{η. We denote by x Þ Ñ E u f pxq ' E s f pxq the associated splitting of T X| U . The line field E u f (resp. E s f ) is everywhere tangent to an f -invariant (singular) holomorphic foliation F u (resp. F s ) coming from the f A invariant linear unstable (resp. stable) foliation on A. Since f is uniformly expanding/contracting on E u{s f , Alternative (c) is not possible. If Alternative (a) holds, then Epxq being f -invariant on a set of full measure, it must coincide with E u f or E s f , say with E u f . By continuity any g P Γ ν preserves E u f pointwise on Supppµq. Since in addition µ is Zariski diffuse, g preserves E u f everywhere on X, so it preserves also the unstable holomorphic foliation F u . From this, we shall contradict the fact that Γ ν is nonelementary. We use a dynamical argument, based on basic constructions which are surveyed in [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF]; one can also derive a contradiction from [START_REF] Cantat | Symétries birationnelles des surfaces feuilletées[END_REF].

Every leaf of F u , except a finite number of them, is parametrized by an injective entire holomorphic curve ϕ : C Ñ X, the image of which is Zariski dense. Fix a Kähler form κ on X and consider the positive currents defined by

(7.4) α Þ Ñ ˜ż R 0 ż Dp0;tq ϕ ˚κ dt t ¸´1 ż R 0 ż Dp0;tq ϕ ˚α dt t
for any smooth p1, 1q-form α. As R goes to `8, it is known that this sequence of currents converges to a closed positive current T f that does not depend on the parametrization ϕ of the leaf, nor on the leaf itself (provided the leaf is Zariski dense). This current is uniquely determined by F u and the normalization xT f |κy " 1. Dynamically, it is the unique closed positive current T f that satisfies xT f |κy " 1 and f ˚T f " λpf qT f for some λpf q ą 1. Its cohomology class rT f s is a non-zero element of H 1,1 pX; Rq of self-intersection 0. Now, pick any element g of Γ ν . Since g preserves F u , it permutes its leaces and preserves the ray R `rT f s. Thus, Γ ν preserves an isotropic line for the intersection form in H 1,1 pX; Rq, and this contradicts the non-elementarity assumption (see [19, §2.3]).

Finally, if alternative (b) holds, any g P Γ ν preserves tE u f pxq, E s f pxqu on a set of full measure so, since µ is Zariski diffuse, it must either preserve or swap these directions. Passing to an index 2 subgroup both directions are preserved, and we again contradict the non-elementary assumption, as in case (a). 7.4. Proof of Theorem 7.3. Let us consider a random dynamical system pX, νq and an ergodic stationary measure µ, as in Theorem 7.3. We keep the notation from §7.1. In particular, we fix a Kähler form κ 0 on X and compute norms with respect to it.

We say that a sequence of real numbers pu n q ně0 almost converges towards `8 if for every K P R, the set L K " tn P N ; u n ď Ku has an asymptotic lower density (7.5)

denspL K q :" lim inf nÑ`8 ˆ7pL K X r0, nsq n `1
ẇhich is equal to 0: denspL K q " 0 for all K.

Lemma 7.5. The set of points x " pω, xq in X `such that D x f n ω almost converges towards `8 on PpT x M q is F `-invariant. In particular, by ergodicity, (a) either D x f n ω almost converges towards `8 for pν N ˆµq-almost every pω, xq; (b) or, for pν N ˆµq-almost every pω, xq, there is a sequence pn i q with positive lower density along which D x f n i ω is bounded.

The proof is straightforward and left to the reader (see [START_REF] Barrientos | Extremal exponents of random products of conservative diffeomorphisms[END_REF]). We are now ready for the proof of Theorem 7.3. Let us first emphasize one delicate issue: in Conclusion (1) of the theorem, it is important that the directions E (resp. E 1 and E 2 ) only depend on x P X (and not on x " px, ωq P X `). Likewise in Conclusion (2), the trivialization P x should depend only on x. This justifies the inclusion of a detailed proof of Theorem 7.3, since in the slightly different setting of [START_REF] Barrientos | Extremal exponents of random products of conservative diffeomorphisms[END_REF], the authors did not have to check this property.

We fix a measurable trivialization P : T X Ñ X ˆC2 , given by linear isometries P x : T x X Ñ C 2 , where T x X is endowed with the hermitian form pκ 0 q x , and C 2 with its standard hermitian form. This trivialization conjugates the action of DF `to that of a cocycle A : X `ˆC 2 Ñ X `ˆC 2 over F `. We denote by A x : tx u ˆC2 Ñ tF `px qu ˆC2 the induced linear map; since A x " P f 1 ω pxq ˝pDf 1 ω q x ˝P ´1

x , we see that A x " A pω,xq depends only on x and on the first coordinate f 1 ω " f 0 of ω. Using P we transport the measure μ to a measure, still denoted by μ, on the product space X ˆP1 pCq. By our invariance assumption, its disintegrations μx " μx satisfy pPA x q › μx " μF `px q " μf 1 ω pxq . The essentially bounded case.-In this paragraph we show that in case (b) of Lemma 7.5, Conclusion (2) of Theorem 7.3 holds. We streamline the argument of [START_REF] Barrientos | Extremal exponents of random products of conservative diffeomorphisms[END_REF]Prop. 4.7] which deals with the more general case of GLpd, Rq-cocycles (see also [START_REF] Arnold | Jordan normal form for linear cocycles[END_REF][START_REF] Zimmer | On the cohomology of ergodic group actions[END_REF]).

Set G " PGLp2, Cq, and define the G-extension r F `of F `on X `ˆG by (7.6) r F `px , gq " pF `px q, PpA x qgq " ppσpωq, f 1 ω pxqq, PpA pω,xq qgq for every x " pω, xq in X `and g in G; thus r F `is given by F `on X `and is the multiplication by PpA x q on G. Since PpA pω,xq q depends on ω only through its first coordinate, r F `is the skew product map associated to the random dynamical system pf pxq, P f 1 ω pxq ˝pDf 1 ω q x ˝P ´1 x q on X ˆG. Denote by P the convolution operator associated to this random dynamical system. Let Prob µ pX ˆGq be the set of probability measures on X ˆG projecting to µ under the natural map X ˆG Ñ X. Since µ is stationary, P maps Prob µ pX ˆGq to itself.

By assumption there is a set E of positive measure in X `, a compact subset K G of G, and a positive real number ε 0 such that (7.7) dens

! n ; PpA pnq x q P K G ) ě ε 0 for all x in E.
Lemma 7.6. There exists an ergodic, stationary, Borel probability measure r µ G on X ˆG with marginal measure µ on X.

Proof. (See [3, Prop. 4.13] for details). Let r µ G be any cluster value of the sequence of probability measures 1 N ř N ´1

i"0 P i pµ ˆδ1 G q. By the boundedness assumption, r µ G has mass M ě ε 0 (the possible escape of mass to 8 in G is not total) and is stationary (i.e. P-invariant). Its projection on the first factor is equal to M µ. We renormalize it to get a probability measure. Then, using the ergodic decomposition and the ergodicity of µ, we may replace it by an ergodic stationary measure in Prob µ pX ˆGq.

Denote by r

m G " ν N ˆr µ G the r F `-invariant measure associated to r µ G . The action of r F `on X `ˆG (resp. of the induced random dynamical system on X ˆG) commutes to the action of G by right multiplication, i.e. to the diffeomorphisms R h defined by (7.8) R h px , gq " px , ghq for h P G. Slightly abusing notation we also denote by R h the analogous map on X ˆG. The next lemma combines classical arguments due to Furstenberg and Zimmer.

Lemma 7.7. Let r µ G be a Borel stationary measure on X ˆG with marginal µ on X. Set

H " th P G ; pR h q › r µ G " r µ G u " th P G ; pR h q › r m G " r m G u .
Then H is a compact subgroup of G and there is a measurable function Q : X Ñ G such that the cocycle B x " Q ´1 f 1 ω pxq ¨PpA x q ¨Qx takes its values in H for pν N ˆµq-almost every x .

Proof. Clearly, H is a closed subgroup of G. If H were not bounded then, given any compact subset C of G, we could find a sequence ph n q of elements of H such that the subsets R hn pCq are pairwise disjoint. Choosing C such that X ˆC has positive r µ G -measure, we would get a contradiction with the finiteness of r µ G . So H is a compact subgroup of G.

We say that a point px, gq in X ˆG is generic if for ν N -almost every ω, (7.9)

1 N N ´1 ÿ n"0 ϕ ´r F n `pω, x, gq ¯ÝÑ N Ñ8 ż X `ˆG ϕ d r m G
for every compactly supported continuous function ϕ on X `ˆG. The Birkhoff ergodic theorem provides a Borel set E of generic points of full r µ G -measure. Now if px, g 1 q and px, g 2 q belong to E, writing g 2 " g 1 h " R h pg 1 q for h " g ´1 1 g 2 , we get that h is an element of H. Given g P G, define E x Ă G to be the set of elements g P G such that px, gq is generic. Then there exists a measurable section

X Q x Þ Ñ Q x P G such that Q x P E x for almost all x.
By definition of E x , pω, x, Q x q satisfies (7.9) for ν N -almost every ω. The r F `-invariance of the set of Birkhoff generic points implies that pf 1 ω pxq, PpA x qQ x q belongs to E for ν-almost every

f 0 " f 1 ω . Since pf 1 ω pxq, Q f 1 ω pxq q belongs to E as well, it follows that Q ´1 f 1 ω pxq PpA x qQ x is in H. We conclude that the cocycle B x " Q ´1 f 1
ω pxq ¨PpA x q ¨Qx takes its values in H for almost all x , as claimed.

Note that the map x Þ Ñ Q x lifts to a measurable map x Þ Ñ Q 1

x P GL 2 pCq. Conjugating H to a subgroup of PU 2 by some element g 0 P G, the two previous lemmas give: if D x f n ω is essentially bounded, then Conclusion (2) of Theorem 7.3 holds (the P x are obtained by composing the Q 1

x with a lift of g 0 to GL 2 pCq). The unbounded case. -Now, we suppose that D x f n ω is essentially unbounded (alternative (a) of Lemma 7.5) and we adapt the results of [3, §4.1] to the complex setting to arrive at one of the Conclusions (1.a) or (1.b) of Theorem 7.3. The main step of the proof is the following lemma.

Lemma 7.8. Let A be a measurable GLp2, Cq-cocycle over pX `, F `, ν N ˆµq admitting a projectively invariant family of probability measures pμ x q xPX such that almost surely A pnq x almost converges to infinity. Then for almost every x, μx possesses an atom of mass at least 1{2; more precisely:

either μx has a unique atom rwpxqs of mass ě 1{2, that depends measurably on x P X; -or μx has a unique pair of atoms of mass 1{2, and this (unordered) pair depends measurably on x P X.

For the moment, we take this result for granted and proceed with the proof. By ergodicity of µ, the number of atoms of μx and the list of their masses are constant on a set of full measure. A first possibility is that μx is almost surely the single point mass δ rwpxqs ; this corresponds to (1.a). A second possibility is that μx is the sum of two point masses of mass 1{2; this corresponds to (1.b). In the remaining cases, there is exactly one atom of mass 1{2 ď α ă 1 at a point rwpxqs. Changing the trivialization P x , we can suppose that rwpxqs " rws " r1 : 0s. Then we write μx " αδ r1:0s `μ 1

x , and apply Lemma 7.8 to the family of measures μ1

x (after normalization to get a probability measure). We deduce that almost surely μ1

x admits an atom of mass ě p1 ´αq{2. Two cases may occur:

-μ1

x has a unique atom of mass β ě p1 ´αq{2, -μ1

x has two atoms of mass p1 ´αq{2.

The second one is impossible, because changing the trivialization, we would have μx " αδ r1:0s 1´α

2 pδ r´1:1s `δr1:1s q, and the invariance of the finite set tr1 : 0s, r´1 : 1s, r1 : 1su would imply that the cocycle PpA x q stays in a finite subgroup of PGL 2 pCq, contradicting the unboundedness assumption.

If μ1

x has a unique atom of mass β ě p1 ´αq{2, we change P x to put it at r0 : 1s (the trivialization P x is not an isometry anymore). We repeat the argument with μx " αδ r1:0s βδ r0:1s `μ 2

x . If β " 1 ´α, i.e. μ2

x " 0, then we are done. Otherwise μ2

x has one or two atoms of mass γ ě p1 ´α ´βq{2, and we change P x to assume that one of them is r1 : 1s and the second one -provided it exists-is rτ pxq : 1s; here, x Þ Ñ τ pxq is a complex valued measurable function. Endow the projective line P 1 pCq with the coordinate rz : 1s; then PpA x q is of the form z Þ Ñ apx qz. Since PpA x q pt1, τ pxquq " pt1, τ pF `px qquq, we infer that:

-either apx q1 " 1 and PpA x q is the identity; -or apx q1 " τ pπ X pF `px qqq and apx qτ pxq " 1 in which case τ pπ X pF `px qqq " τ pxq ´1.

Thus we see that along the orbit of x , apF n `px qq takes at most two values τ pπ X pF n `px qqq ˘1, and A pnq x is bounded, which is contradictory. This concludes the proof.

Proof of Lemma 7.8. Let r and ε be small positive real numbers. Let Prob r,ε pP 1 pCqq be the set of probability measures m on P 1 pCq such that sup xPP 1 mpBpx, rqq ď 1{2 ´ε, where the ball is with respect to some fixed Fubini-Study metric. This is a compact subset of the space of probability measures on P 1 . The set

(7.10) G r,ε " γ P PGLp2, Cq, Dm 1 , m 2 P Prob r,ε pP 1 pCqq, γ › m 1 " m 2 (
is a bounded subset of PGLp2, Cq. Indeed otherwise there would be an unbounded sequence γ n together with sequences pm 1,n q and pm 2,n q in Prob r,ε pP 1 pCqq such that pγ n q › m 1,n " m 2,n . Denote by γ n " k n a n k 1 n the KAK decomposition of γ n in PGLp2, Cq, with k n and k 1 n two isometries for the Fubini-Study metric; since γ n is unbounded, we can extract a subsequence such that the measures pk 1 n q › m 1,n and pk ´1 n q › m 2,n converge in Prob r,ε pP 1 pCqq to two measures m 1 and m 2 , while the diagonal transformations a n converge locally uniformly on P 1 pCqz tr0 : 1su to the constant map γ : P 1 pCqz tr0 : 1su Þ Ñ tr1 : 0su. Then (7.11) γ › `m1|P 1 pCqztr0:1su ˘" m 1 pP 1 pCqz tr0 : 1suqδ a ď m 2 ;

since m 1 belongs to Prob r,ε pP 1 pCqq, m 1 pP 1 pCqz tr0 : 1suq ě 1{2`ε, hence m 2 ě p1{2`εqδ a , in contradiction with m 2 P Prob r,ε pP 1 pCqq. This proves that G r,ε is bounded.

To prove the lemma, let us consider the ergodic dynamical system PDF `, and the family of conditional probability measures μx for the projection pω, x, vq Þ Ñ x " pω, xq. If there exist r, ε ą 0 such that μx belongs to Prob r,ε pP 1 pCqq for x in some positive measure subset B then, by ergodicity, for almost every x P X `there exists a set of integers Lpx q of positive density such that for n P Lpx q, F n `px q belongs to B, hence A pnq x belongs to G r,ε ( 5 ). From the above claim we deduce that A pnq x is uniformly bounded for n P Lpx q, a contradiction. Therefore for every r, ε ą 0, the measure of x , μx P Prob r,ε pP 1 pCqq ( is equal to 0; it follows that for almost every x , μx possesses an atom of mass at least 1{2.

If there is a unique atom of mass ě 1{2, this atom determines a measurable map x Þ Ñ rwpx qs P PT x X; since μx does not depend on ω, rwpx qs depends only on x, not on ω. If there are generically two atoms of mass ě 1{2, then both of them has mass 1{2, and the pair of points determined by these atoms depends only on x.

CHARACTERIZATION OF UNIFORM EXPANSION

In this section we build on the previous results, in conjunction with the measure rigidity results from our previous work [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF], to find sufficient conditions for as well as obstructions to uniform expansion for a non-elementary action on a compact complex surface. (i) either both Lyapunov exponents of µ 1 are non-positive, (ii) or µ 1 is hyperbolic and its field of Oseledets stable directions is non-random. Theorem 5.5 asserts that the existence of non-expanding ν-stationary measures is the obstruction to uniform expansion of ν: Corollary 8.2 (of Theorem 5.5). Let X be a compact complex surface and ν be a probability measure on the group AutpXq, satisfying the moment condition (M). Then ν is uniformly expanding if and only if non-expanding ν-stationary measures do not exist, hence if and only if every ergodic ν-stationary measure µ on X satisfies one of the following properties:

µ has a positive Lyapunov exponent and its stable distribution depends non-trivially on the itinerary; -the two Lyapunov exponents of µ are strictly positive.

Groups with invariant curves.

Proposition 8.3. Let X be a compact complex surface. Let Γ be a subgroup of AutpXq that preserves a complex curve C Ă X. If ν is a probability measure on Γ satisfying (M), then ν is not uniformly expanding. Remark 8.4. We leave the reader check that the proof adapts to the real case in the following sense: if X, Γ and C are defined over R and CpRq is of dimension 1 (that is, neither empty nor a finite set), then ν is not uniformly expanding in restriction to CpRq. Lemma 8.5. Let C be a compact Riemann surface. Then, AutpCq does not support any uniformly expanding probability measure.

Proof. Let κ be a Kähler form on C that satisfies ş C κ " 1. For every f P AutpCq, Note that the same argument applies to conformal diffeomorphisms, in particular for Diff 1 pS 1 q. Lemma 8.5 and Remark 2.1 imply Proposition 8.3 when C is smooth; we now prove Proposition 8.3 in full generality.

ş C f ˚κ " 1 " ş C }D x f } 2 κ " 1,
Proof of Proposition 8.3. Arguing by contradiction, we assume that ν is uniformly expanding. Let Γ 1 ď Γ be the finite index subgroup fixing each component of C, and each of its branches at each of its singular points; let ν 1 be the hitting measure on Γ 1 associated to ν pn 0 q , where n 0 is as in Equation (2.6). By Proposition 3.2, ν 1 is uniformly expanding, so by replacing ν by ν 1 and C by one of its components we assume now that C is irreducible and all branches at its singular points are fixed by Γ. To get a contradiction we will construct a stationary measure µ supported on C such that the tangential Lyapunov exponent along T C is non-positive.

By Lemma 8.5 we may assume that the singular set SingpCq is non-empty. If the genus of C is ě 0, the invariance of SingpCq forces Γ| C to be finite, in contradiction with the uniform expansion of ν. Thus, C is a rational curve; let π : Ĉ Ñ C be its normalization and Γ Ă Autp Ĉq » PGL 2 pCq be the group induced by Γ; the measure ν induces a measure ν on Γ. Fix p P Ĉ such that p :" πppq is singular. The germ of curve given by Ĉ at p determines one of the branches of C at p; our assumptions imply that p is fixed by Γ. There are local coordinates t P pC, 0q for p Ĉ, pq and pz, wq P pC 2 , p0, 0qq for pX, pq in which π is expressed as a Puiseux expansion (8.1) t Þ Ñ pπ 1 ptq, π 2 ptqq " pαt q , βt r q modulo higher order terms where 1 ď q ă r; if q " 1 the branch is smooth at p. In these coordinates, the tangent direction to C at p corresponding to the branch determined by p is given by p1, 0q P C 2 . Let λ p Ĉ,pq be the Lyapunov exponent of ν at p, and λ pC,pq be the Lyapunov exponent of ν in the tangent direction of this branch.

Lemma 8.6. With notation as above λ pC,pq " qλ p Ĉ,pq . In particular λ pC,pq and λ p Ĉ,pq have the same sign.

Proof. Pick f P Γ, write f pz, wq " pf 1 pz, wq, f 2 pz, wqq in the local coordinates pz, wq, and expand f 1 in power series: f 1 pz, wq " ř i,j a i,j z i w j . Since the branch determined by λ p Ĉ,pq is f -invariant, we have D p f p1, 0q " pa 1,0 , 0q with a 1,0 ‰ 0. Thus, (8.2) f 1 pπptqq " 8 ÿ i,j"0 a i,j α i β j t qi`rj " a 1,0 αt q mod pt q`1 q. Now, f lifts to an automorphism f of Ĉ fixing p. Writing f ptq " λt mod pt 2 q, we get π 1 p f ptqq " αλ q t q mod pt q`1 q. Then, the semiconjugacy f 1 pπptqq " π 1 p f ptqq gives λ q " a 1,0 , and we are done.

We resume the proof of Proposition 8.3. We fix an affine coordinate s on Ĉ » P 1 pCq such that p " 8. Then, every lift ĝ P Γ can be written as an affine map ĝpsq " a g s `bg . Proof. For the spherical metric, the derivative of ĝ at 8 in Ĉ is 1{a g . The computations of Lemma 8.6 show that the derivative of g acting on X in the direction of the branch of C at πp8q is 1{a q g for some q ě 1. So (M) implies that Ep|log |a g ||q ă 8. Since ν is uniformly expanding, this direction is repelling on average: by Lemma 8.6, we get Eplog ˇˇa ´1 j ˇˇq ą 0. To estimate |b g |, we note that dist X pπpsq, pq -|s| ´q when s P C approaches 8. Changing the affine coordinate s if necessary, we may assume that πp0q ‰ p. We get

(8.3) 1 |b g | q -dist X pπpĝp0qq, πp8qq " dist X pgpπp0qq, gppqq ď }g} C 1 dist X pπp0q, pq.
From this and (M) it follows that Eplog `|b g |q ă 8.

The integrability provided by Lemma 8.7 now allows us to construct a stationary measure with full mass in the affine chart C Ă C with non-positive Lyapunov exponent (relative to the affine metric). This is classical, we briefly recall the argument for completeness (see [START_REF] Brandt | The stochastic equation Yn`1 " AnYn `Bn with stationary coefficients[END_REF]). For ω " pg n q ně0 , write g n psq " a n s `bn , and consider the sequence of right products r n pωq " g 0 ¨¨¨g n´1 . One easily checks that (8.4) r n pωqpsq " a 0 ¨¨¨a n´1 s `n´1 ÿ j"0 a 0 ¨¨¨a j´1 b j .

For ν N -almost every ω, 1 n log |a 0 ¨¨¨a n´1 | converges to λ :" Eplog |a g |q ă 0. Fix ε ă |λ|. Since Eplog `|b g |q ă 8, ř 8 j"0 νt|b g | ą e εj u ă 8. By the Borel-Cantelli Lemma, |b j | ď e εj for ν N -almost every ω and for large j; hence, the series on the right hand side of (8.4) converges. It follows that r n pωqpsq converges almost surely to a limit e ω that does not depend on s P C. The distribution of e ω is the desired stationary measure µ C . If µ is any stationary measure with µpCq " 1, then r n pωq › µ converges to δ eω almost surely: this shows that µ C is the unique stationary measure with µpCq " 1; in particular, µ C is ergodic. Since the affine derivative of g is the constant a g , the Lyapunov exponent of µ C , relative to the affine metric, is equal to λ.

To conclude the proof, note that µ :" π › pµ C q is an ergodic ν-stationary measure on X which has a well-defined Lyapunov exponent, thanks to the moment condition (M). If µ gives positive mass to the singular set of C, then it must be concentrated on a single singular point of C (and likewise µ C is a single atom in Ĉ). By Lemma 8.6 the corresponding branch is attracting on average, which contradicts uniform expansion. Therefore µ gives no mass to SingpCq, and we claim that its Lyapunov exponent λpµq| T C in the direction of C equals λ (even if the ratio between the ambient and affine metrics on C Ă C is unbounded). Indeed, for µˆν N -almost every px, ωq and v P T 1

x C, we can fix a subsequence n j such that f n j ω pxq is far from the singularities of C (hence from p " πp8q). If j is large,

1 n j log › › D x f n j ω
› › is both close to λ and to λpπ › µq| T C . We conclude that λpπ › µq| T C ă 0, which again is contradictory. The proof is complete. 8.1.3. Zariski diffuse measures. From now on we focus on the case of a minimal Kähler surface X of Kodaira dimension zero, that is, a torus, a K3 surface, or an Enriques surface. In this case AutpXq preserves a canonical volume form vol X (see Example 1.4).

From Corollary 8.2, the obstruction to uniform expansion is the existence of a non-expanding stationary measure µ. Moreover, in the first case of Definition 8.1, both exponents must vanish because we are in a volume preserving setting. In this situation, Theorems 7.2 and 7.3 give a precise description of µ.

Theorem 8.8. Let X be a torus, a K3 surface, or an Enriques surface. Let ν be a probability measure on AutpXq satisfying (M) such that Γ ν is non-elementary. If µ is a Zariski diffuse ν-stationary measure, the following properties are equivalent (a) µ is non-expanding; (b) the fiber entropy h µ pX, νq vanishes.

Morover under these assumptions, µ is invariant and h µ pf q " 0 for every f P Γ ν .

Proof. As a preliminary step, observe that almost every ergodic component of µ is Zariski diffuse: this follows from the fact that there are only finitely many invariant curves and countably many isolated periodic points. In addition, by linearity of the entropy, if h µ pX, νq " 0 then almost every ergodic component of µ has zero fiber entropy as well. Thus for both implications we may further assume that µ is ergodic as a stationary measure.

Since there is an invariant volume form, either both Lyapunov exponents of µ vanish or µ is hyperbolic. In the first case, the invariance principle guarantees that µ is Γ ν -invariant and the fibered version of the Ruelle inequality (see e.g. [19, §7]) implies that its fiber entropy vanishes. If µ is hyperbolic, the invariance of µ and the vanishing of the entropy follow from [19, Thm. 9.1]. Thus, Property (a) implies Property (b), together with the invariance of µ.

Consider the converse implication. Again, if µ has zero Lyapunov exponents then it is nonexpanding and invariant. Otherwise it is hyperbolic and by applying the whole argument of [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF] in the complex case, we infer that if the stable directions of µ depend on the itinerary, its conditionals along Pesin unstable manifolds admit a non-trivial translation invariance; in particular they are non-atomic. It follows that h µ pX, νq ą 0 (see also [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]Cor. 9.2]). So under assumption (b) the stable directions are non-random and, as already explained, µ is invariant by [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]Thm. 9.1].

The fact that h µ pf q " 0 for all f P Γ ν will be shown in Theorem B.1.

8.1.4. Refined criterion. The discussion of the previous paragraphs leads to a version of Theorem 1.5 that does not require Γ ν to contain parabolic elements:

Theorem 8.9. Let X be a compact Kähler surface which is not rational. Let ν be a probability measure on AutpXq satisfying (M) such that Γ ν is non-elementary. Then ν is uniformly expanding if and only if the three following conditions hold:

(1) every finite Γ ν -orbit is uniformly expanding;

(2) there is no Γ ν -invariant algebraic curve;

(3) there is no Zariski diffuse invariant measure µ with zero fiber entropy.

Proof. If a compact Kähler surface X is ruled (over a curve of positive genus) or has a positive Kodaira dimension, then AutpXq is elementary (in the first case, it preserves the ruling; in the second case, it preserves the Kodaira-Iitaka fibration, acting as a finite group on the base). Thus, the Kodaira dimension of X vanishes. If X is not minimal, the uniqueness of the minimal model shows that there is a AutpXq-invariant curve, and we know this is incompatible with uniform expansion (Proposition 8.3). Now if kodpXq " 0, X is minimal, and AutpXq is non-elementary, then X is a torus, a K3 surface, or an Enriques surface; hence, we can assume that X is such a surface.

If ν is uniformly expanding, Property ( 1) is obvious, Property (2) follows from Proposition 8.3, and Property (3) follows from Corollary 8.2 and Theorem 8.8.

Conversely, if these properties hold, and if µ is an ergodic ν-stationary measure then by Property (2) µ is either Zariski diffuse or finitely supported. Then, Theorem 8.8 and Property (1) imply that µ is not non-expanding, and we conclude with Corollary 8.2.

Proof of Theorem 1.5. This follows directly from Theorem 1.3 and Theorem 8.9.

Remark 8.10. The proof of (b)ñ(a) in Theorem 8.8 relies on the following fact: for a hyperbolic stationary measure, if the stable directions of µ depend on the itinerary, then its unstable conditionals satisfy some non-trivial translation invariance. This is the "easy part" of the adaptation of [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF] to complex surfaces; the "difficult part" would be to obtain some SRB property from this invariance (either on X or on some totally real surface associated to the stationary measure). We did not provide a proof for this fact because the arguments of [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF] can be applied directly. As a consequence, this fact is also used in the implication "uniformly expanding implies (3)" in Theorem 8.9. On the other hand, it is not used in Theorem 1.5 because in this case the condition (3) of Theorem 8.9 is automatically satisfied, thanks to Theorem 1.3; it is not used either for the part of Theorem 8.9 asserting that the assumptions (1), ( 2) and (3) imply uniform expansion.

Remark 8.11. Using Theorem 7.4 instead of Theorem 1.3 gives a version of Theorem 1.5 where the existence of a parabolic element in Γ is replaced by the existence of a Kummer element. The details of the adaptation are left to the interested reader.

As already said, it should be expected that vol X be the only invariant measure for a typical non-elementary action on a surface X of Kodaira dimension zero; similarly, vol Y should be the unique invariant measure on Y , if Y Ă X is an invariant totally real surface. This is why we plug this property as a hypothesis in the next corollary. Corollary 8.12. Let X be an irrational compact Kähler surface. Let ν be a probability measure on AutpXq such that (a) ν satisfies Condition (M), (b) Γ ν is non-elementary, and (c) there is no Γ ν -invariant curve. Let Y Ă X be a Γ ν -invariant totally real surface. Assume that (d) the action of Γ ν on H 1 pY ; Rq has a positive Lyapunov exponent, and (e) the only Γ ν -invariant measure on Y is the induced volume vol Y . Then ν is uniformly expanding on Y .

Proof. Under these assumptions there is no finite Γ ν -orbit. The invariance of vol Y , Assumption (d), and Corollary 5.3 show the existence of an ergodic, hyperbolic, ν-stationary measure µ on Y . By [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]Thm. 10.10], µ must be Γ ν -invariant, so by Assumption (e), µ " vol Y . Arguing as in § 7.2.2, the hyperbolicity of vol Y as a stationary measure implies that its fiber entropy is positive. Thus vol Y is hyperbolic and has positive fiber entropy. Now, assume that ν is not uniformly expanding on Y . By Theorem 5.5, there is a nonexpanding stationary measure µ 1 on Y , which must be Zariski diffuse since there is no proper Zariski closed subset. By Theorem 8.8 µ 1 is invariant -so again µ 1 " vol Y -and its fiber entropy vanishes. This is a contradiction, and the proof is complete. Remark 8.13. If the stiffness result of [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF] were known to hold on X, we would get the same result on X (instead of on a totally real surface Y Ă X). More precisely, assume (a), (b), (c) and (e'): the only Γ ν -invariant ergodic probability measures on X are vol X or vol Y for some Γ ν -invariant totally real surface Y Ă X. Then we would conclude that ν is uniformly expanding on X.

8.2.

Uniform expansion along finite orbits. Using classical results on random products of matrices, it is easy to characterize when a fixed point under Γ ν is uniformly expanding. We say that a subgroup of GL 2 pCq is strictly triangular if it is reducible with exactly one invariant direction.

Proposition 8.14. Let X be a torus, a K3 surface, or an Enriques surface. Let ν be a probability measure on AutpXq satisfying (M), and let x 0 be a fixed point of Γ ν . Then ν is uniformly expanding on T x 0 X if and only if one of the following holds (a) the induced action of Γ ν on T x 0 X is non-elementary; (b) this action is strictly triangular and its invariant direction is expanding.

If ν is symmetric, it is uniformly expanding on T x 0 X if and only if (a) holds.

In case (b) there exists u P T x 0 X such that f › u " λ f u for every f P Γ ν , and the expansion means that ş log |λ f | dνpf q ą 0.

Proof (see also [START_REF] Prohaska | Markov random walks on homogeneous spaces and Diophantine approximation on fractals[END_REF]). By Lemma 2.3, to prove uniform expansion it is enough to show that for every v P T x 0 X, lim inf nÑ8 1 n log }pf n ω q › v} ą 0. The proof is based on the work of Furstenberg and Kifer [START_REF] Furstenberg | Random matrix products and measures on projective spaces[END_REF] (see also [10, §3.7]). These references deal with general random products of matrices in GL d pRq; in our volume preserving situation the Lyapunov exponents λ 2 ď λ 1 of the random product in GL 2 pCq satisfy λ 1 `λ2 " 0, so they can be read off directly from the action on PT x 0 X. According to Theorems 3.5 and 3.9 of [START_REF] Furstenberg | Random matrix products and measures on projective spaces[END_REF], there are two possibilities:

(i) for every v P T x 0 X and ν N -almost every ω, 1 n log }pf n ω q › v} Ñ λ 1 ; (ii) there exists a non-random, Γ ν -invariant filtration t0u " L 2 ă L 1 ă L 0 " T x 0 X and β 1 ă β 0 such that for i " 0, 1 for any v P L i zL i`1 , for ν N -almost every ω, 1 n log }pf n ω q › v} Ñ β i . Furthermore β 0 " λ 1 .
We now compare this dichotomy with the classification of subgroups of PGL 2 pCq (with a slight abuse of notation, we also denote by Γ ν the induced subgroup of PGL 2 pCq). ´If Γ ν is strongly irreducible, we are in case (i) and there are two possibilities. If Γ ν is proximal (hence non-elementary) then λ 1 ą 0 and ν is uniformly expanding. If Γ ν is not proximal, it is contained in a compact subgroup and ν is not uniformly expanding.

´If Γ ν is irreducible but not strongly irreducible, we are in case (i) and there are two lines which are permuted by Γ. In some affine coordinate z on PT x 0 X, Γ ν is conjugate to a subgroup of tz Þ Ñ λz ε ; λ P C ˆ, ε " ˘1u and ε " ´1 with positive probability. In this case λ 1 " 0 (see e.g. [START_REF] Dujardin | Degenerations of SLp2, Cq representations and Lyapunov exponents[END_REF]Prop. 5.3]), so ν is not uniformly expanding.

´If Γ ν is reducible it preserves one or two directions in T x 0 X. If Γ preserves a direction with exponent ď 0, then ν is not uniformly expanding. So, we can assume that Γ preserves a unique direction, and that the corresponding exponent β is positive. By (i) and (ii) we see that lim nÑ8 1 n log }pf n ω q › v} ě β for any v P T x 0 X and almost every ω; so ν is uniformly expanding. This covers all possible cases and the proof is complete.

Let F be a finite set, viewed as a 0-dimensional manifold, and V be a real or complex vector bundle of dimension d over F ; identify V with F ˆKd , for K " R or C. Let GLpV q be the group of bijections of V acting linearly on fibers: it is a semidirect product GLpV q » SpF q ĠL d pKq F where SpF q acts on GL d pKq F by permuting the factors. We say that a subgroup of GLpV q is strongly irreducible if it acts transitively on F and the stabilizer of any x P F acts strongly irreducibly on the fiber txu ˆKd of V ; equivalently, if there is no invariant and finite collection of subspaces of dimension ‰ 0, d in some fibers of V . Similar notations and notions are defined for PGLpV q.

Assume now that F is a finite Γ-orbit on X, and consider the induced action of Γ on T X| F :" Ť xPF T x X. We say that this action is non-elementary if its image in PGL pT X| F q is strongly irreducible and unbounded. When Γ preserves a volume form on X, its image in GL pT X| F q is unbounded if and only if it is unbounded in PGL pT Xq. We say that it is strictly triangular if the only proper Γ-invariant subbundle in T X| F is given by a 1-dimensional subbundle L Ă T X| F .

Pick a point x in F and set Γ x " Stab Γ ptxuq. Since F is an orbit, rΓ : Γ x s " |F | and the image of Γ in PGL pT X| F q is unbounded if and only if the image of Γ x in PGL pT x Xq is unbounded. Thus, one easily gets the following lemma. Lemma 8.15. If F is a finite Γ-orbit, the action of Γ on T X| F is non-elementary (resp. strictly triangular) if and only if for some, hence any, x P F the action of Stab Γ ptxuq on T x X is non-elementary (resp. strictly triangular). Theorem 8.16. Let X be a torus, a K3 surface, or an Enriques surface. Let ν be a probability measure on AutpXq satisfying (M `), and F be a finite Γ ν -orbit. Then ν is uniformly expanding on F if and only if the induced action of Γ ν on T F is (a) either non-elementary; (b) or strictly triangular and the field of invariant directions L Ă T X| F is uniformly expanding.

If ν is symmetric, it is uniformly expanding on F if and only if (a) holds.

Proof. Let Γ F be the finite index subgroup fixing every point of F . Assume that ν is uniformly expanding. Then by Proposition 3.2, for some n 0 , the induced measure pν pn 0 q q Γ F is uniformly expanding. Therefore, by Proposition 8.14 Γ F satisfies Property (a) or (b) at every point of F , and we conclude by Lemma 8.15. Conversely, assume that (a) or (b) holds. Note that by Theorem 3.1, ν Γ F satisfies (M `). By Lemma 8.15 and Proposition 8.14, ν Γ F is uniformly expanding on F , hence by Proposition 3.3 ν is uniformly expanding on F , as desired.

This theorem shows that when ν is symmetric all conditions in Theorem 8.9 depend only on Γ ν , and not on ν. Thus we obtain: Corollary 8.17. Let X be a torus, a K3 surface, or an Enriques surface. Let Γ be a nonelementary subgroup of AutpXq. Let ν and ν 1 be symmetric probability measures on AutpXq satisfying (M `) such that Γ ν " Γ ν 1 " Γ. Then ν is uniformly expanding if and only if ν 1 is uniformly expanding.

In the following if X is a torus, K3, or Enriques surface, we will say that the action of a non-elementary subgroup Γ Ă AutpXq is uniformly expanding if this property holds for some (hence any) symmetric probability measure ν satisfying (M `) and generating Γ.

EXAMPLES OF UNIFORMLY EXPANDING ACTIONS

9.1. A finitary version of Theorem 1.5 and application to Wehler surfaces. In [22, § §7-8], Chung uses computer assistance to prove the uniform expansion of some concrete algebraic actions on real surfaces. In our situation Theorem 1.5 can be used to check uniform expansion, but this requires a description of all invariant Zariski-closed subsets. As already explained, invariant curves can be determined by cohomological computations; for instance, if X is a generic Wehler surface, there is no AutpXq-invariant curve. Thus the main problem is to study finite orbits.

If the group Γ is non-elementary, contains parabolic elements, and has no invariant curve, the main result of [START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF] says that Γ admits only finitely many finite orbits, except when pX, Γq is a Kummer example. However, the proof given in [START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF] does not provide any bound on the number or the lengths of such orbits; so, there is a priori no hope of numerically checking uniform expansion along all of them, nor proving that there are no finite orbits. The next result explains how to overcome this issue. Theorem 9.1. Let X be a smooth projective surface and Γ be a non-elementary subgroup of AutpXq containing parabolic elements, which does not preserve any algebraic curve. Assume that we are given: (i) algebraic equations for X, and the formulas defining a generating subset S of Γ; (ii) a basis of NSpX; Rq and the matrices of s › : NSpX; Rq Ñ NSpX; Rq, for s in S; (iii) a parabolic element g P Γ, given as a word in the generators s P S, and its invariant fibration π : X Ñ B.

Then, there is an analytically computable integer N pX, Γq such that the action of Γ on any finite orbit of length greater than N pX, Γq is non-elementary.

By analytically computable, we mean computable by a computer able to solve real analytic equations; by algebraically computable, we mean computable by a computer able to solve algebraic equations. The proof will provide an analytically computable subset containing all possible non-expanding finite orbits.

Example 9.2. Let h P Γ be a conjugate of g with a distinct invariant fibration. Denote by Tor N pgq the finite set of fibers of the g-invariant fibration in which g is a periodic translation of period ď N . Then, the set of finite orbits of Γ of length ď N is algebraically computable since it is contained in (9.1)

Tor N pgq X Tor N phq " tx P X ; g N pxq " h N pxq " xu.

A typical application of Theorem 9.1 is to the Wehler family. Recall from § 1.1 that W 0 is the family of Wehler surfaces which are smooth and do not contain any fiber of the three natural projections pP 1 q 3 Ñ pP 1 q 2 . Under these assumptions the group Γ generated by the three basic involutions σ 1 , σ 2 and σ 3 is non-elementary and has no invariant curve (see [START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF]Prop. 2.2]). It turns out that in this case N pX, Γq is constant on a Zariski dense open subset (see Proposition 9.7 below). This leads to: Theorem 9.3. There is a dense Zariski open subset of W 0 (resp. of the family W 0 pRq of real Wehler surfaces), in which the action of Γ " xσ 1 , σ 2 , σ 3 y is uniformly expanding on X. 9.1.1. Preliminaries on Halphen twists. Let us resume the discussion from § 6.1 and add a few preliminaries on Betti foliations and the non-twisting locus. Let h be a Halphen twist with associated fibration π : X Ñ B. Consider a simply connected open subset U of B ˝together with a section σ : U Ñ X of π and a continuous frame for the homology of the fibers above U . For w P U , one can identify the fiber X w to C{Latpwq (σpwq corresponding to the zero of C{Latpwq), as in § 6.1. Then, above U , there is a unique real-analytic diffeomorphism Ψ : π ´1pU q Ñ U ˆR2 {Z 2 such that (a) π ˝Ψ " Ψ ˝πU , where π U is the projection onto U ; (b) Ψ maps σ to the zero section w Þ Ñ pw, p0, 0qq of π U , and maps the basis of H 1 pX w ; Zq to the standard basis of H 1 pR 2 {Z 2 ; Zq " Z 2 ; (c) on each fiber, Ψ is a real analytic isomorphism of real Lie group.

Above U , the Betti foliation is the foliation by submanifolds of the form Ψ ´1pU ˆtpx, yquq; these leaves are local holomorphic sections of π, with σ corresponding to Ψ ´1pU ˆtp0, 0quq. Conjugating by Ψ, we get

(9.2) Ψ ˝h ˝Ψ´1 : pw, px, yqq Þ Ñ pw, px, yq `T pwqq,
where T : U Ñ R 2 {Z 2 is real analytic. By [20, Lem. 3.9], the map T is an (orientation preserving) branched covering, so it behaves topologically like w Þ Ñ w k . In U , the non-twisting locus NT h is the set tw P U ; D w T " 0u; equivalently, NT h X U " πptt 1 , . . . , t q uq, where tt 1 , . . . , t q u is the set of tangencies between the Betti foliation and the section h ˝σ. These definitions do not depend on the above choices and NT h can indeed be defined globally on B ˝.

A key fact is that NT h is a finite subset of B ˝(see [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF]Prop. 3.14] or [START_REF] Duistermaat | Discrete integrable systems. QRT maps and elliptic surfaces[END_REF]Cor. 7.7.10]). We denote by |NT h | its cardinality, and by multpNT h q its cardinality counted with multiplicity, that is, taking into account the degree of the local branched covering T .

Note that, once h and π are given, the set NT h Ă B ˝is analytically computable: one has to compute the periods of X w to get Latpwq, then Ψ is R-linear from C{Latpwq to R 2 {Z 2 , and T is then obtain from h by conjugacy. 9.1.2. Proof of Theorem 9.1. As in [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF], for pg, hq P HalpΓq 2 we set (9.3) STangpπ g , π h q " Singpπ g q Y Singpπ h q Y Tang tt pπ g , π h q,

where Singpπ g q is the union of all singular and multiple fibers, and Tang tt pπ g , π h q is the part of the tangency locus of π g and π h which is not contained in Singpπ g q Y Singpπ h q. Put NT X g " π ´1

g pNT g q (so that NT X g is a curve in X) and likewise NT X h " π ´1 h pNT h q. Lemma 9.4. Let g, h be a pair of Halphen twists in Γ with distinct invariant fibrations, and let x P X be a point with a finite Γ-orbit. If this orbit is not uniformly expanding, then it is contained in STangpπ g , π h q Y NT X g Y NT X h .

Proof. We argue by contraposition: replacing x by another point in its orbit if necessary, we assume that x R STangpπ g , π h q Y NT X g Y NT X h , and we want to show that its orbit is uniformly expanding. Since Γpxq is finite, there are positive integers k and such that g k and h are in Stab Γ pxq. By definition of the non-twisting locus, g k and h induce parabolic homographies on PpT x Xq; and since x R Tang tt pπ g , π h q, the fixed points of these homographies are distinct; thus, the action of xg k , h y on PpT x Xq is non-elementary. By Proposition 8.14 and Lemma 8.15, the orbit of x is uniformly expanding.

The intersection number of NT X g Y STangpπ g , π h q (resp. NT X h Y STangpπ g , π h q) with a smooth fiber X h w (resp. X g w ) does not depend on the fiber. Let n 0 pg, hq be the maximum of these intersection numbers: (9.4) n 0 " maxt rNT X g Y STangpπ g , π h qs ¨rX h w s ; rNT X h Y STangpπ g , π h s ¨rX g w squ. The set STangpπ g , π h q can be computed algebraically, thus (9.5) n 0 pg, hq ď Apg, hq maxp|NT g | ; |NT h |q `Bpg, hq

where Apg, hq and Bpg, hq can be computed algebraically (by computing the tangency loci and intersection numbers). Then, we set (9.6) npg, hq " n 0 pg, hq! Lemma 9.5. Let g, h be a pair of Halphen twists in Γ with distinct invariant fibrations. Let x P X be such that Γpxq is finite and not uniformly expanding. Then Γpxq is contained in

STangpπ g , π h q Y pNT X g X NT X h q Y pNT X g X NT X h n gh ´n q Y pNT X h X NT X
g n hg ´n q, where n " npg, hq is defined by (9.6).

Proof. The statement of the lemma concerns the orbit Γpxq, but we only have to prove it for x itself. If x P STangpπ g , π h q Y pNT X g X NT X h q we are done. Otherwise by Lemma 9.4, x belongs to NT X g zNT X h or NT X h zNT X g . Assume that x P NT X g zNT X h . The h-orbit of x is finite and by Lemma 9.4 again, for every q, h q pxq is contained in X h

x X pNT X g Y STangpπ g , π h qq (here we abuse notation and write X h

x for X h π h pxq ). Thus, h n pxq " x, where n " npg, hq. Set f " h n gh ´n. The fiber X f x associated to f through x is h n pX g x q, and since x R NT X h , X f

x is transverse to X g x at x, as well as to X h x . Moreover, x belongs to NT X f , because x belongs to NT X g and h n pxq " x. Hence x P NT X g X NT X h n gh ´n . Doing the same in the case where x P NT X h zNT X g completes the proof.

The set NT g is analytically computable (by § 9.1.1), and Critpπ g q is algebraically computable. Similarly, if h is in HalpΓq, Tang tt pπ g , π h q X NT X f is analytically computable. The previous lemma shows that all non uniformly expanding finite orbits are contained in Badpg, hq :" STangpπ g , π h q Y pNT X g X NT X h q Y pNT X g X NT X h n gh ´n q Y pNT X h X NT X g n hg ´n q for every pair pg, hq P HalpΓq 2 with distinct invariant fibrations, where n " npg, hq as in Equation (9.6). Intersecting these sets for various choices of pg, hq, we expect to get a finite analytically computable set. Observe that Badpg, hq is the union of STangpπ g , π h q and a finite set, because NT X g X NT X h is finite when π g and π h are distinct. So, what remains to do is to exhibit an explicit finite set of pairs pg, hq such that the intersection of the STangpπ g , π h q is finite. We first treat the case of Wehler surfaces, which is sufficient to proceed with Theorem 9.3.

Conclusion of the proof of Theorem 9.1 in the Wehler case. Fix a Wehler surface X P W 0 and consider the three pairs pg 1 , g 2 q, pg 2 , g 3 q, pg 3 , g 1 q, where g 1 " σ 2 ˝σ3 , g 2 " σ 3 ˝σ1 and g 3 " σ 1 ˝σ2 . Note that the g i -invariant fibration is the i-th projection π i .

Assume that the intersection of the divisors STangpπ i , π j q contains an irreducible curve D Ă X. If D is contained in Singpπ i q X Singpπ j q with i ‰ j, then pπ i , π j q maps D onto a point and this contradicts the fact that X P W 0 . If D is contained in, say, Tang tt pπ 1 , π 2 q and Tang tt pπ 2 , π 3 q, the three fibrations are pairwise tangent along D, and we obtain a contradiction because there is no tangent vector v ‰ 0 to pP 1 q 3 which is mapped to 0 by each Dπ i . The last possibility is that D is contained in, say, Tang tt pπ 1 , π 2 q and Singpπ 3 q. In this case, there is a point p on D at which D p π 3 : T p X Ñ T π 3 ppq P 1 is equal to 0, and at such a point, the same contradiction applies. This shows that (9.7)

F pg 1 , g 2 , g 3 q :" Badpg 1 , g 2 q X Badpg 2 , g 3 q X Badpg 3 , g 1 q is finite, with an analytically computable cardinality, and the proof is complete.

Remark 9.6. The above proof provides a computation of the integer N pXq involving:

(1) algebraic quantities that are constant on W 0 , like STangpπ i , π j q ¨STangpπ j , π k q, (2)

|NT g i | for i " 1, 2, 3.
Therefore, if |NT g i | ď B, then N pXq ď N pBq for some N pBq depending only on B. Indeed, the number n in Lemma 9.5 depends only on n 0 (see Equations (9.6)) and by Equation (9.4)) n 0 is bounded by a function of B. Then, because the norm of pg n 0 i q ˚: NSpXq Ñ NSpXq is bounded by Cn 2 0 for some uniform constant C, we obtain NT X g i X NT X g n j g i g ´n j ď C 1 n 2 0 B 2 for some constant C 1 and the result follows.

Conclusion of the proof of Theorem 9.1 in the general case. By assumption, Γ is non-elementary and has no invariant curve. Let Γ ˚be its image in GLpNSpX; Zqq.

If g is the parabolic element given in assumption (iii) of the theorem, up to sign, there is a unique integral primitive class cpgq P NSpXq such that g › cpgq " cpgq and cpgq ¨cpgq " 0. By the assumptions (ii) and (iii), this class can be computed explicitly. An element f of AutpXq preserves the g-invariant fibration π (permuting its fibers) if and only if it fixes cpgq. Since Γ is non-elementary, it is not contained in the stabilizer of cpgq. Thus, according to Proposition 3.2 of [START_REF] Eskin | On uniform exponential growth for linear groups[END_REF], there is a computable integer N , and a composition f of length N in the generators s P S that does not preserve cpgq. Then, h :" f ˝g ˝f ´1 is a parabolic element of Γ with invariant fibration π ˝f ‰ π.

Since g and its invariant fibration π, as well as f , are explicit, we can compute the degree of STangpπ g , π h q (for the embedding X Ă P m pCq given by assumption (i)). Denote by pC i q iPI the irreducible components of STangpπ g , π h q; we have |I| ď degpSTangpπ g , π h qq.

Suppose that for each C i , one can exhibit some f i P Γ for which f i pC i q Ć STangpπ g , π h q. Then the set of pairs

(9.8) tpg, hqu Y pf i gf ´1 i , f i hf ´1 i q ; i P I ( satisfies | Ş i STangpπ g i , π h i q| ă `8
, and we are done because the cardinality of this finite set is algebraically computable. So, we now fix such an irreducible component C i , and we construct such an f i .

First, assume that C i is an irreducible component of Tang tt pπ g , π h q. Then C i is generically transverse to π, hence degpg n pC i qq tends to infinity. We can thus set f i " g n i for some large enough n i (explicitely computable from the action on NSpXq).

The second case is when C i is an irreducible component of Singpπ g qYSingpπ h q; in particular, its self-intersection C 2 i is ď 0. By [START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF]Thm D], there exists a loxodromic element f 0 P Γ without invariant curve; in particular f |I|! 0 pC i q ‰ C i . Since f 0 is loxodromic, this inequation is equivalent to pf |I|! 0 q › rC i s ‰ rC i s. Indeed, either C 2 i " 0 and we readily get a contradiction since a loxodromic element does not fix any non-zero isotropic class, or C 2 i ă 0 and this follows from f |I|! 0 pC i q ‰ C i since rC i s determines C i when the self-intersection is negative. Thus, if we set (9.9) W i :"

! f P GLpNSpX, Rqq : f |I|! › rC i s " rC i s ) ,
we see that Γ ˚is not contained in W i . Proposition 3.2 of [START_REF] Eskin | On uniform exponential growth for linear groups[END_REF] then provides a computable element f P Γ such that f R W i . Now, if f q pC i q were contained in STangpπ g , π h q for 0 ď q ď |I|, we would find two integers q 1 ă q 2 ď |I| such that f q 2 ´q1 pC i q " C i ; in particular, f |I|! pC i q would be equal to C i , a contradiction. Thus, there is an iterate f i :" f q i , with q i ď |I|, such that f i pC i q Ć STangpπ g , π h q, and the proof is complete. 9.1.3. Proof of Theorem 9.3. Recall that, for Wehler surfaces, Γ " xσ 1 , σ 2 , σ 3 y.

Proposition 9.7. There exists an analytically computable integer N such that for any Wehler surface X P W 0 , any finite Γ-orbit of length ą N is non-elementary.

This uniform bound is the main step towards Theorem 9.3. In view of Remark 9.6, this proposition follows from Theorem 9.1 and the following uniformity result. Proposition 9.8. For any g P tg 1 , g 2 , g 3 u, the cardinality of NT g is uniformly bounded in W 0 .

Let X Ă W 0 ˆpP 1 ˆP1 ˆP1 q be the universal family of Wehler surfaces, as in [18, §2]. As X varies in W 0 , the automorphisms g i and their invariant fibrations π i depend on X, but for notational simplicity we drop the dependence in X.

From now on, we fix g P tg 1 , g 2 , g 3 u; its invariant fibration π : X Ñ P 1 is the restriction of one of the projections π i to X; its base does not depend on X. Lemma 9.9. Let X 0 P W 0 , w 0 P NT g , and k be the multiplicity of w 0 in NT g . Let U Ă P 1 be a topological disk such that U X NT g " tw 0 u and U X Critpπq " H. Then, there exists a neighborhood V of X 0 in W 0 such that for any X in V , the total multiplicity of NT g in U is equal to k.

Proof. Fix an open connected neighborhood V of X 0 such that for X in V , -U does not intersect any of the sets Critpπq; -there is a section w Þ Ñ ς X pwq of X Ñ W 0 ˆP1 above V ˆU , together with a continuous choice of basis for the homology of the fibers of π above U .

Then the sections, the Betti foliations (above U ), and their lifts to U ˆC all depend continuously on X in V . In particular, we can find a disk U 1 Ă P 1 , with w 0 P U 1 Ť U , whose boundary is a smooth Jordan curve γ, and such that for any X P V , the Betti foliation is transverse to g ˝ςX above γ. In particular, NT g is disjoint from γ. Now, recall that the map T defined in Equation (9.2) behaves topologically like w Þ Ñ w k`1 ; in such a local coordinate, k `1 is the winding number of the curve T ˝γ around T p0q. Since NT g stays disjoint from γ for X in V , this winding number is constant in V ; thus, the number of points of NT g enclosed by γ (counted with multiplicity) stays constant on V . The lemma follows.

Lemma 9.10. There exists a proper semi-algebraic subvariety Z g Ă W 0 of positive codimension such that multpNT g q is locally constant in W 0 zZ g .

Proof.

Step 1: Keeping away from the singular fibers. Fix X 0 P W 0 and w 0 P X 0 a critical value of π. It is shown in [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF]Lem. 3.11] that NT g does not accumulate w 0 . Here, we show that outside a semi-algebraic subvariety Z g Ă W 0 this non-accumulation holds uniformly with respect to X: we shall construct a neighborhood V ˆU of pX 0 , w 0 q such that U is disjoint from NT g for every X in V . For this, we review the proof of [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF]Lem. 3.11] and make it locally uniform in X under appropriate hypotheses on X 0 .

Define W 1 to be the dense, Zariski open subset ( 6 ) of W 1 such that for any X P W 1 and any i P t1, 2, 3u, all singular fibers of π i are of type I 1 . In this case there are 24 such fibers (the 6 To show that W1 is dense, we only have to show that it is non-empty. This is a consequence of the following fact. Let X be in W0, let π1 : X Ñ P 1 be the first projection, and let m be a critical point of π1. Let F be the fiber of π1 containing m. Then, each of the conditions (1) the singularity of F at m is degenerate (in the sense of Morse, i.e. it is not a A1-singularity);

(2) F contains a second singular point m 1 defines a proper subset of W0. In other words, these properties (1) and (2) disappear after a generic small perturbation of X in W0, which can be checked directly.

Euler characteristic of a K3 surface is 24, the contribution to the Euler characteristic of a smooth fiber is 0, and the contribution of an I 1 fiber is 1). Suppose that X 0 P W 1 .

Fix a small disk U Ă P 1 centered at w 0 and containing no other singular value of π : X 0 Ñ P 1 . Fix a neighborhood V of X 0 in W 1 , and local coordinates on U (depending on X), so that (i) this property persists for X P V and (ii) the unique singular value of π in U is w 0 " 0. Let X # U be the complement in X g U :" π ´1pU q of the unique singular point of X g w 0 . We fix a reference section ς X : U Ñ X # U depending holomorphically on X P V and w P U . For X P V and w P U ztw 0 u we can write X g w » C{Z ' Zτ X pwq, as in § 6.1. Since the singular fiber X g w 0 is of type I 1 and w 0 " 0, the monodromy along a simple loop around 0 maps the basis p1, τ X pwqq to p1, τ X pwq `1q. Moreover, X # U is biholomorphic to the quotient of U ˆC by the family of lattices Z ' Zτ X pwq, where τ X pwq " 1 2iπ logpk X pwqq for a function k X : U Ñ C which has a single zero at the origin and depends holomorphically on X P V and w P U . Since g ˝ςX is another section of π above U , there is a holomorphic function t X pwq of X and w such that the lift of g to U ˆC is given by pw, zq Þ Ñ pw, z `tX pwqq. The calculations of [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF] (see §3.3.2 and Lemma 3.11 there) show that the equation for NT g in U is (9.10) ´i logp|k X pwq|qk X pwq t 1 X pwq " k 1 X pwqImpt X pwqq. We claim that if Impt X p0qq ‰ 0, then by reducing V and U if necessary, NT g X X g

U " H. Indeed if U is small enough, there exist positive constants ε, c such that for any X P V , (9.11) |k X pwq| ď ε, ˇˇk 1 X pwq ˇˇě c, |Impt X pwqq| ě c, and ˇˇt 1 X pwq ˇˇď c ´1. Reducing U further, ε can be chosen arbitrary small while c remains bounded away from 0. If ε log ε ă c 3 , this is not compatible with the equality (9.10), so NT g X U " H. Lemma 9.11. The locus (9.12) tX P V : Impt X pw 0 qq " 0u is a semi-algebraic subset of positive codimension.

Proof of Lemma 9.11. Consider the Wehler surfaces X Ă V Ă W 1 , and their equations (9.13) A 222 x 2 y 2 z 2 `A221 x 2 y 2 z `¨¨¨`A 100 x `A010 y `A001 z `A000 " 0.

Permuting coordinates if necessary, we suppose that π : X Ñ P 1 is the projection onto the first coordinate. As X varies near X 0 , the critical value of π near w 0 and the corresponding critical point in X can be computed algebraically in terms of the A ijk . Using the action of PGLp2, Cq 3 on P 1 ˆP1 ˆP1 , we may assume that w 0 " 0 (as above) and the unique singular point of the fiber X g w 0 :" X X tx " 0u is p0, 0q. So, the equation of

X g w 0 in P 1 ˆP1 is (9.14) ay 2 z 2 `by 2 z `cyz 2 `dyz `ey 2 `f z 2 " 0,
for some coefficients a, . . . , f given by algebraic expressions in the A ijk . Since X P W 1 , X g w 0 has two transverse branches at p0, 0q: their tangent directions are given by the solutions of dyz `ey 2 `f z 2 " 0 in P 1 .

One can also write X g 0 zt0, 0u as the quotient of t0u ˆC by the lattice Latp0q " Z; in this coordinate, g acts as multiplication by expp2iπt X p0qq. Thus, Impt X p0qq " 0 means that g induces a rotation, instead of a loxodromic homography, on the rational curve X g 0 . Writing down g " σ y ˝σz in coordinates, we obtain (9.15) gpy, zq "

˜´1 ´d2

ef d e ´d f ´1¸ˆy z ˙`O ´}py, zq} 2 for py, zq P X g 0 . Thus, D p0,0q g P GLpT 0,0 Xq has determinant 1 and trace ´2 ´d2 ef . As a consequence, g acts as a rotation on X g 0 if and only if 2 `d2 ef P r´2, 2s: this is a semi-algebraic condition.

To conclude, we let W g Ă W 0 be the intersection of W 1 with the complement of the subsets tX P W 1 ; Impt X pw i q " 0u for each of the 24 singular values w i of π. We finally define Z g to be the complement of W g ; by Lemma 9.11, it is a proper semi-algebraic set of positive codimension.

Step 2: Conclusion. Pick X 0 P W 0 zZ g and cover P 1 by a finite family F of topological disks, such that for every U P F , U contains at most one point of Critpπq Y NT g . If U P F contains a critical value of π (and no point of NT g ), then, as already explained, this property persists in a neighborhood of X 0 . By Step 1, for X sufficiently close to X 0 , U is disjoint from NT g as well.

For the remaining disks, the local constancy of multpNT g q follows from Lemma 9.9. The proof is complete.

Proof of Proposition 9.8. We use a semi-continuity argument. Since the exceptional set Z g defined in Lemma 9.10 is semi-algebraic, the open set W 0 zZ g is also semi-algebraic, so it admits finitely many connected components (see [START_REF] Bierstone | Semianalytic and subanalytic sets[END_REF]Cor. 2.7] for instance). Thus, by Lemma 9.10, multpNT g q and therefore |NT g | are uniformly bounded on W 0 zZ g , say |NT g | ď B. Now, pick X 0 P Z g (thus X 0 P W 0 ) and assume that for X 0 one has |NT g | ą B. We can then consider a finite number of small topological disks U i with disjoint closures in P 1 , such that |NT g | X Ť U i ą B. By Lemma 9.9, these non-twisting points persist for X close enough to X 0 . Since W 0 zZ g is dense in W 0 , this contradicts the definition of B and the proof is complete.

Proof of Theorem 9.3. The main point of [18, Thm A] is that the set of X P W 0 possessing a finite orbit of length ď B is a proper Zariski closed subset Z B of W 0 . For N as in Proposition 9.7, for any X P W 0 zZ N , all finite orbits of Γ are uniformly expanding. We conclude by applying Theorem 1.5 (with ν " 1 3 pδ σ 1 `δσ 2 `δσ 3 q). The proof of the corresponding statement in W 0 pRq is identical. Remark 9.12. We expect that an analogue of Theorem 9.3 holds for other families with large automorphism groups containing parabolic elements, like Enriques surfaces, or the family associated to pentagon folding (see [19, §3]).

Remark 9.13. The proof of Proposition 9.8 suggests that there should exist a notion of multiplicity, including singular fibers, for which multpNT g q would be constant on W 0 and would be an algebraically computable invariant of the parabolic automorphism g. A variant of this question is mentioned in [START_REF] Duistermaat | Discrete integrable systems. QRT maps and elliptic surfaces[END_REF]Rmk 7.7.4]. 9.2. Thin subgroups. In this section we consider the total space W of all Wehler surfaces and the universal family X Ă W ˆpP 1 ˆP1 ˆP1 q. We change a little bit the notation: Γ will be a subgroup of Z{2Z ˚Z{2Z ˚Z{2Z, and Γ X will be the corresponding subgroup of AutpXq.

Let E be an elliptic curve. Consider the following classical Kummer construction (see [18, §4]): let η be the involution ηpx, yq Þ Ñ p´x, ´yq on A :" E ˆE; the associated Kummer surface is the desingularization { E ˆE{η; the natural GLp2, Zq action on E ˆE descends to E ˆE{η and induces a non-elementary automorphism group of { E ˆE{η. The surface E Ê{η can be realized as a singular Wehler example (see [16, §8.2]); in addition the action of Z{2Z ˚Z{2Z ˚Z{2Z is induced by a finite index subgroup of GLp2, Zq. Let us briefly recall the construction: write E in Weierstrass form y 2 " 4x 3 ´g2 x ´g3 , with the neutral element of the group law on E located at infinity. To pm 1 , m 2 q P E ˆE, m i " px i , y i q, we associate m 3 " ´pm 1 `m2 q and φpm 1 , m 2 q " px 1 , x 2 , x 3 q, where m 3 " px 3 , y 3 q. Then, φ is η-invariant and determines a biregular map φ : E ˆE{η Ñ X E onto a singular Wehler surface X E with 16 nodal singularities.

Assume that Γ Ă Z{2Z ˚Z{2Z ˚Z{2Z is not virtually cyclic. Then for X P W 0 , Γ X is non-elementary (see [19, §3.1]). Theorem 9.14. Let Γ be a subgroup of Z{2Z ˚Z{2Z ˚Z{2Z which is not virtually cyclic. For X P W 0 sufficiently close to X E , the subgroup Γ X is uniformly expanding on X.

Thus for every "abstract" non-elementary subgroup Γ of Z{2Z˚Z{2Z˚Z{2Z, the open subset W exp pΓq of those X P W 0 for which the action of Γ X is uniformly expanding is non-empty. The group Γ can be arbitrarily thin, in particular it is not assumed to contain parabolic elements. In view of Theorem 8.9, it is natural to expect that W exp pΓq is actually dense.

Proof. The difficulty is that we cannot directly argue that uniform expansion is an open property, because X E is singular. Lemma 9.15. Fix f P Z{2Z ˚Z{2Z ˚Z{2Z, and denote also by f the induced fibered map on the universal family of p2, 2, 2q-surfaces in pP 1 q 3 . Then f is regular on a neighborhood of X E . Proof of Lemma 9.15. Pick a p2, 2, 2q surface X. If X does not contain any fiber of the projection π 12 " pπ 1 , π 2 q : pP 1 q 3 Ñ pP 1 q 2 , then the same property holds in a neighborhood V of X of the universal family of p2, 2, 2q-surfaces, and σ 3 determines an automorphism of V. Thus, we only have to prove that X E does not contain any fiber of the projections π ij . Let us show that X E does not contain any vertical line tx " x 0 , y " y 0 u. Such a line would provide a family of points pm 1 , m 2 q on E ˆE with fixed first coordinates x 1 " x 0 , x 2 " y 0 , for which the first coordinate of m 3 :" ´pm 1 `m2 q takes arbitrary values. This is impossible. The same argument applies to the lines ty " y 0 , z " z 0 u and tz " z 0 , x " x 0 u because the relation m 1 `m2 `m3 " 0 is symmetric (equivalently, the equation of X E given in [16, §8.2] is symmetric in px, y, zq).

There is a finite index subgroup of Γ that fixes each singularity of X E . By Proposition 3.3 and the fact that uniform expansion does not depend on the measure, we can replace Γ with this finite index subgroup, endow Γ with a finitely supported, symmetric measure ν with Γ " Γ ν , and then we have to prove that pΓ X , ν X q is uniformly expanding for X P W 0 near X E ; here, ν X is the measure induced by ν on Γ X .

Endow P 1 ˆP1 ˆP1 with the Fubini study metric, and the Wehler surfaces X with the induced metric. Recall that T 1 X denotes the unit tangent bundle.

Assume, by way of contradiction, that there is a sequence X n Ñ X E along which ν Xn is not uniformly expanding. For each n, let denote the natural projection T 1 X n Ñ X n . For X E , denote by T 1 X E the subset of T 1 pP 1 ˆP1 ˆP1 q which coincides with T 1 RegpX E q above the regular part of X E and coincides with T 1

x pP 1 ˆP1 ˆP1 q above each singularity x P SingpX E q; again, we denote by the projection T 1 X E Ñ X E . The proof of Theorem 5.1 provides a sequence of stationary measures μXn on T 1 X n (with projections µ Xn :" ˚μ Xn ) such that (9.16) ż log }f › u}dν Xn pf q dμ Xn puq ď 0.

From Lemma 9.15, we can extract a subsequence, still denoted by pX n q, such that pμ tn q converges to a stationary measure μX E on T 1 X E satisfying (9.17)

ż log }f › u}dν X E pf q dμ X E puq ď 0.
By iterating and using the stationarity of μX E , the same inequality holds with ν pmq X E instead of ν X E for every m ą 0. To get the desired contradiction, we shall show that no such measure exists.

Step 1: near the singularities.-Here we show that there exists n 0 P N, c 0 ą 0, and an open neighborhood U of SingpX E q such that if u P T 1 X E and puq P U , then (9.18)

ż log }f › u} dν pn 0 q X E pf q ě c 0 .
By continuity it is enough to prove this when puq P SingpX E q. Recall that Γ X E fixes SingpX E q pointwise. Around each of its singularities, X E is locally isomorphic to the quotient C 2 {η, ηpu, vq " p´u, ´vq, standardly embedded in C

'.

Thus, the action of Γ X E on the tangent cone T 0 X E is, up to a linear conjugacy, given by φ › pG 0 q. This is a subgroup of Opq; Rq » O 2,1 pRq, where q is the quadratic form qpx, y, zq " xz ´y2 .

By assumption, it is a non-elementary group of isometries of q, hence it acts strongly irreducibly and proximally on R 3 Ă C 3 (loxodromic elements of GL 2 pZq are mapped to loxodromic elements in Opq; Rq). It preserves the real decomposition C 3 " R 3 ' R iR 3 and the action on R 3 and iR 3 are linearly conjugate (by multiplication by i). Therefore, as in § 8.2, the Inequality (9.18) follows from [START_REF] Furstenberg | Random matrix products and measures on projective spaces[END_REF] (see also [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF], Chap. III, Cor. 3.4(iii)).

Step 2: away from the singularities.-We shall show that there exists a neighborhood U 1 Ă U of SingpX E q and c ą 0 such that for any fixed u P T 1 X E such that puq R U 1 , (9.21)

P ˆ1 m log }pf m ω q › u} ě c ˙ÝÑ mÑ8 1.
By Lemma 2.3 (see also Remark 2.4), this implies that E plog }pf m ω q › u}q ě mc{2 for large m. Then, the first step and a compactness argument identical to that of Lemma 2.2 show that uniform expansion holds on T 1 X E , which is the desired contradiction.

Let U 1 be an open neighborhood of SingpX E q which will be specified later. There is a constant δ " δpU 1 q such that (9.22) if puq R U 1 and pf › uq R U 1 , then log }f › u} ě log }f › u} flat ´δ,

where }¨} flat is the Riemannian metric on RegpX E q induced by the flat metric of E ˆE.

The pull-back of ν to GLp2, Zq generates G 0 and its support is finite. Since G 0 Ă GLp2, Zq is non-elementary, we have uniform expansion with respect to the flat metric. By Lemma 2.3, there exists a constant c 1 ą 0 and sets of trajectories

Ω 1 m Ă Ω such that PpΩ 1 m q Ñ 1 as m Ñ 8 and (9.23) if ω P Ω 1 m , 1 m log }pf m ω q › u} flat ě c 1 .
Fix ε ą 0 and 0 ă c ă c 1 . By the first step of the proof, SingpX E q is repelling on average in P 1 ˆP1 ˆP1 ; so there is a Margulis function on X E with poles at SingpX E q. Thus, there is an open set U 1 " U 1 pεq Ă U with the following property: for large enough m, the set Ω 2 m of trajectories ω P Ω such that pf m ω qp puqq R U 1 satisfies PpΩ 2 m q ě 1 ´ε{2. Now, U 1 being fixed, for m large enough we have that PpΩ 1 m XΩ 2 m q ě 1´ε and by (9.22) and (9.23), if

ω P Ω 1 m XΩ 2 m , (9.24) 1 m log }pf m ω q › u} ě c 1 ´δpU 1 q m ě c.
Thus, the convergence (9.21) holds and the proof is complete.

10. APPLICATIONS 10.1. Orbit closures. The following is a version of the orbit closure Theorem E of [START_REF] Cantat | Invariant measures for large automorphism groups of projective surfaces[END_REF] in which periodic orbits are allowed. Combined with Theorem 9.3, it gives Theorem 1.1.

Theorem 10.1. Let X be a torus, a K3 surface, or an Enriques surface. Let Γ Ă AutpXq be a non-elementary subgroup which contains parabolic elements and does not preserve any algebraic curve. Assume that for any finite orbit O, the induced action of Γ on T X| O is nonelementary. Then there exists a finite set F and a real analytic, totally real, and Γ-invariant surface Y Ă X with SingpY q Ă F such that for every x P X:

(a) either x belongs to F (and its orbit is finite); (b) or x belongs to Y zF and Γpxq is a union of components of Y ;

(c) or Γpxq " X.

Proof. First observe that under these assumptions, [START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF]Thm C] implies that there exists a maximal finite invariant subset F . Fix a symmetric measure ν such that Γ ν " Γ and satisfying the moment condition (M `). By Theorems 1.5 and 8.16, ν is uniformly expanding. We now resume the discussion from [20, §8], in particular Remark 8.6 there: STang Γ is a finite invariant set and if x P X is such that Γpxq is infinite but not dense, then there are two possible situations:

(1) either ΓpxqzSTang Γ is discrete outside STang Γ ;

(2) or ΓpxqzSTang Γ ": Y pxq is a totally real analytic surface, whose singular locus SingpY q is discrete outside STang Γ .

In Case (1), Γpxq is finite. Indeed Γpxq is at most countable, so if µ is any cluster value of 1 n ř n´1 k"0 ν k ‹ δ x , then µ is a purely atomic stationary measure. In this case it follows from Theorem 4.3 that the orbit of x must be finite, hence contained in F .

If Case (2) holds, we first claim that SingpY pxqq is finite. Indeed, SingpY pxqq is a Γ-invariant countable set, which clusters only at STang Γ . By the previous argument, every orbit Γpyq in SingpY pxqq is finite, so by the finiteness of the set of finite orbits [18, Thm C] we conclude that SingpY pxqq itself is finite. Now, let µ 1 be a cluster value of 1 n ř n´1 k"0 ν k ‹ δ x . By Theorem 4.3, µ 1 is an atomless stationary measure supported on Y pxq such that µ 1 pRegpY pxqqq " 1. Since Γ has no invariant curve, µ 1 is Zariski diffuse. Let µ be any ergodic component of µ 1 . Theorems 8.8 and 1.3 imply that µ is hyperbolic and its stable directions depend genuinely on the itinerary. Then the argument of [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF]Thm 3.1] adapts immediately to show that µ is SRB( 7 ). The canonical invariant 2-form of X induces a Γ-invariant measure vol Y pxq on Y pxq (see Lemma 6.3). Since RegpY pxqq admits a Margulis function, we conclude from Proposition 4.4 that the volume vol Y pxq is finite. Therefore we can copy verbatim the argument of [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF]Thm 3.4] to conclude that µ is Γ-invariant. Since [20, Thm C] says that there are only finitely many Γ-invariant measures, there are only finitely many possible surfaces Y pxq. Taking Y to be their union, the proof is complete. 10.2. Ergodicity. In [START_REF] Dolgopyat | On simultaneous linearization of diffeomorphisms of the sphere[END_REF], the original motivation to introduce uniform expansion was a criterion for ergodicity. The same holds in our setting, with a few caveats which will be explained below.

Theorem 10.2 Cor. 2], see also [START_REF] Liu | Lyapunov exponents approximation, symplectic cocycle deformation and a large deviation theorem[END_REF][START_REF] Chung | Stationary measures and orbit closures of uniformly expanding random dynamical systems on surfaces[END_REF]). Let X be a torus, a K3 surface, or an Enriques surface. Let Γ Ă AutpXq be a non-elementary subgroup with a uniformly expanding action on X. Then vol X is Γ-ergodic.

Likewise, if Y Ă X is a Γ-invariant totally real analytic subset such that Γ acts transitively on the set of irreducible components of Y , then vol Y is Γ-ergodic. 7 We are not claiming that we can extend [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF] to non-compact surfaces here. All the necessary estimates on the Lyapunov norms and Pesin charts hold by viewing µ as a hyperbolic stationary measure on the compact complex manifold X. The only issue appears when considering the size and intersection properties of real stable and unstable manifolds in Y pxq, starting from §9.7 of [START_REF] Brown | Measure rigidity for random dynamics on surfaces and related skew products[END_REF]. At this stage Brown and Rodriguez-Hertz already discard a set of small measure of points with bad properties (see the definition of Λpγ1q on p. 1087); so it is enough to remove from this Λpγ1q the set of small measure of points too close to SingpY pxqq, and proceed with their argument.

Note that the notion of irreducible component in real analytic geometry is not well-behaved in general (see [20, §5.1] for a short discussion). Here we content ourselves with saying that Y is irreducible when RegpY q is connected. Observe also that the ergodicity of vol X follows directly from Theorem 6.4 when Γ contains a parabolic element.

Proof (sketch). The proof in [START_REF] Dolgopyat | On simultaneous linearization of diffeomorphisms of the sphere[END_REF] is a bit sketchy, but it was already expanded in [START_REF] Liu | Lyapunov exponents approximation, symplectic cocycle deformation and a large deviation theorem[END_REF][START_REF] Chung | Stationary measures and orbit closures of uniformly expanding random dynamical systems on surfaces[END_REF] (see also [START_REF] Zhang | On stable transitivity of finitely generated groups of volume-preserving diffeomorphisms[END_REF]). Here we just make a few comments on (1) the extension to the holomorphic case for the action on X, and (2) how to deal with the possible singularities for the action on totally real surfaces Y .

Regarding the action on X, let us recall that the proof of [START_REF] Dolgopyat | On simultaneous linearization of diffeomorphisms of the sphere[END_REF] is a variation on the Hopf argument in which the asymptotic behavior of the Birkhoff sums 1 n ř n´1 k"0 δ f n ω pxq is propagated along chains of local stable manifolds (associated to different ω's), to ultimately conclude that there is a uniform r so that almost every point x is located at distance at least r from the boundary of its ergodic component. The key technical ingredients are the facts that under the uniform expansion assumption:

-stable directions at a given point do not concentrate, more precisely there exists α ą 0 such that for any x P X and any rvs P PpT x Xq, the probability that d P 1 prvs, rE s px, ωqsq ă α is smaller than 1/100: this follows from a compactness argument (see [START_REF] Liu | Lyapunov exponents approximation, symplectic cocycle deformation and a large deviation theorem[END_REF]Prop. 4.4.4]); -the Pesin local stable manifolds have uniformly bounded geometry (e.g. uniformly bounded size in the sense of [19, §7.4]): this follows from the usual proof of the local stable manifold theorem; -the absolute continuity of the local stable foliation in Pesin charts: we can copy the usual proof or notice that in the holomorphic case this follows from the fact that the holonomy of a holomorphic motion is quasiconformal.

Given these facts, we can copy the proof of [START_REF] Dolgopyat | On simultaneous linearization of diffeomorphisms of the sphere[END_REF] by plugging in §10.4 the the following elementary geometric property, whose proof is left to the reader: let w " pw 1 , w 2 q P C 2 with }w} ă 1 (possibly close to 1) and E w be the direction perpendicular to the line p0wq; then if L is a complex line containing w, such that the angle in P 1 between the direction of L and E w is greater than α, then L X Bp0, 1q contains a disk of radius r ě rpαq.

For the second statement of the theorem we can directly resort to [START_REF] Liu | Lyapunov exponents approximation, symplectic cocycle deformation and a large deviation theorem[END_REF][START_REF] Chung | Stationary measures and orbit closures of uniformly expanding random dynamical systems on surfaces[END_REF], except that we have to take into account the possibility of singular points on Y , which affect the size and geometry of local stable manifolds on Y . For this, we may argue exactly as in Theorem 10.1: first, the existence of a Margulis function guarantees that vol Y is finite. Next, since uniform expansion holds on X, the size and angle change of local complex stable manifolds is uniformly controlled. Thus, when restricting to Y , we also have a uniform control of this geometry outside any δ-neighborhood of SingpY q. Since the Hopf argument is local, we get that there is a single ergodic component outside a δ-neighborhood of SingpY q, for every δ ą 0, and we conclude by letting δ tend to zero.

Remark 10.3. The argument of [START_REF] Dolgopyat | On simultaneous linearization of diffeomorphisms of the sphere[END_REF] works for a random dynamical system on a (real) compact 2d-dimensional manifold enjoying a uniform expansion property along d dimensional tangent subspaces. This assumption does not hold in our setting since along a totally real subspace one may witness both expansion and contraction. In particular the complex uniform expansion condition is not stable under C 1 perturbations by (real) volume preserving diffeomorphisms of X. Still, the philosophy of the above proof is that the argument is robust enough so that uniform expansion along complex 1-dimensional tangent subspaces in a 2-dimensional complex surface guarantees ergodicity. 10.3. Equidistribution. In the following results, given an action of pΓ, νq on M we say that that random trajectories from x equidistributes towards µ if for ν N -almost every ω 1 n ř n k"1 δ f k ω Ñ µ, where the convergence is in the weak › topology. By averaging with respect to ν N and applying the Dominated Convergence Theorem, this implies that 1 n ř n k"1 ν k ˚δx Ñ µ as well. The following theorem already appears under stronger moment assumptions in [START_REF] Chung | Stationary measures and orbit closures of uniformly expanding random dynamical systems on surfaces[END_REF]Thm D].

Theorem 10.4. Let X R be a smooth real projective surface and ν a probability measure on AutpX R q satisfying (M `). Assume that Γ ν preserves a smooth volume form vol on XpRq and that ν is uniformly expanding on XpRq. Then for any x P X one of the following alternatives holds:

(a) either Γ ν ¨x is finite; (b) or the random trajectories from x equidistribute towards vol X 1 pRq , the normalized induced volume on a union of components of XpRq.

Recall from Theorem 1.5 that the uniform expansion assumption holds when Γ ν contains parabolic elements, has no invariant curve, and that the induced action of Γ ν on finite orbits is uniformly expanding. In this case by [18, Thm C], the number of finite orbits is finite. By Theorem 9.3 this applies to generic real Wehler surfaces and yields Theorem 1.2.

Proof. The result follows directly from the classification of stationary measures in [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF], the ergodicity Theorem 10.2 , and the existence of a Margulis function (Theorem 4.1).

The next result is conditional to the ν-stiffness property of complex non-elementary uniformly expanding actions. We expect that it will be established in the near future.

Theorem 10.5. Let X be a K3 or Enriques surface and ν be a finitely supported measure on AutpXq. Assume that (1) Γ ν is non-elementary, contains parabolic elements, has no invariant curve, and every finite Γ-orbit is uniformly expanding; (2) ν-stiffness holds, that is, every ν-stationary measure is invariant.

Then there exists a finite set F and a (possibly singular) totally real analytic surface Y such that for every x P X: line fields, or (3) none of the above. In cases (1) and (2) we conclude that Γ is elementary by adapting the argument of [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]Thm. 9.1]: this does not rely on the additional real structure. In case (3), since µ is hyperbolic for f , Theorems 7.2 and 7.3 imply that µ is hyperbolic as a stationary measure and as in the proof of Theorem 8.8 we deduce that h µ pX, νq ą 0, which is contradictory. Thus, case (3) does not happen, and we deduce that Γ is elementary for every g P Γ ν , which is a contradiction. Therefore h µ pf q " 0.

What remains to do is to adapt this argument to the case where µ is not ergodic under f . So consider f P Γ ν and assume that h µ pf q ą 0 so that f is loxodromic. As before there are 3 cases: either (1) there is a Γ ν -invariant measurable line field, or (2) there is a Γ ν -invariant pair of measurable line fields, or (3) none of ( 1) and [START_REF] Baouendi | Real submanifolds in complex space and their mappings[END_REF]. We first observe that as before Case 3 does not happen: indeed if there is no invariant line field or pair of invariant line fields, by Theorems 7.2 and 7.3, either µ is hyperbolic as a ν-stationary measure, or the projectivized tangent action of Γ ν reduces to a compact subgroup. But since h µ pf q ą 0, f admits non-zero Lyapunov exponents on a set of positive measure so the latter is impossible. Hence µ is hyperbolic as a ν-stationary measure, and since there is no invariant line field, stable directions depend on the itinerary and as before we conclude that h µ pX, νq ą 0, a contradiction. So one of Cases (1) or (2) holds.

So assume there exists a measurable Γ ν -invariant line field x Þ Ñ rEpxqs P PpT x Xq and pick g P Γ ν . Assume further that g is loxodromic. We will derive a contradiction by showing that xf, gy must be elementary: this is a contradiction because any non-elementary subgroup of AutpXq contains a purely loxodromic non-elementary subgroup. Let P be the measurable partition into ergodic components (under f ) and denote by µ P the conditional measure on P P P, so that that µ " ş µ Ppxq dµpxq is the ergodic decomposition of µ. Since the entropy function is affine, there exists a f -invariant set B of positive measure such that for any x P B, h µ Ppxq pf q ą 0. In particular f is non-uniformly hyperbolic along B, so along B, E must coincide almost everywhere with one of E s f or E u f . Reducing B to a smaller invariant subset we may assume that E " E s f almost everywhere along B. For every n P Z, the automorphism g ´nf g n is loxodromic, preserves µ, is non-uniformly hyperbolic along g ´npBq, and E coincides with E s g ´nf g n almost everywhere By measure preservation there exists m ‰ n such that µpg ´npBq X g ´mpBqq ą 0, so µpB X g m´n pBqq ą 0. Letting h " g m´n f g ´pm´nq and A " B X g m´n pBq we are exactly in the situation of Lemma 11.2 of [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF], and we conclude that W s pf, xq " W s ph, xq for µ-almost every x P A, from which it follows that T f " T h and finally pg m´n q ˚T f " cT f . Since g is loxodromic, this implies that T f " T g or T f " T ǵ , and finally that xf, gy is elementary, which is the sought-after contradiction.

Finally, if there is a measurable pair tE 1 , E 2 u of line fields which is ν-a.s. invariant, we get a f -invariant set B of positive measure along which tE 1 pxq, E 2 pxqu "

! E s f pxq " E u f pxq
) , and a set A " B Xg m´n pBq of positive measure along which

! E s f pxq " E u f pxq
) " E s h pxq " E u h pxq, where h " g m´n f g ´pm´nq , and we conclude as before.

T H pωq ω .

 ω Define the k-th hitting time T H,k of H by T H,1 " T H and the induction T H,k`1 pωq " min tn ě T H,k `1 ; f n ω P Hu. The convolution ν pkq H describes the distribution of f T H,k pωq ω

Proposition 4 . 4 .

 44 Under the assumptions of Theorem 4.3, any stationary Radon measure on M zF has finite mass.

3 7. 3 .

 33 A variant of Theorem 1.3. Let us first recall the definition of classical Kummer examples

8. 1 .

 1 Proof of Theorem 1.5 and related results.

8. 1 . 1 .

 11 Applying Chung's criterion. Definition 8.1. Let ν be a probability measure on AutpXq. A ν-stationary measure µ on X is said non-expanding if every ergodic component µ 1 of µ satisfies:

  so by the Jensen inequality ş C log }D x f }κ ď 0. Now, if ν is any probability measure on AutpCq, then ż C ż AutpCq log }D x f }dνpf qκ ď 0, hence Property (2.6) cannot be satisfied by ν (for any n 0 ě 1).

Lemma 8 . 7 .

 87 The functions log |a g | and log `|b g | are ν-integrable and Eplog |a g |q ă 0.

  (a) either x belongs to F ; (b) or x belongs to Y zF and its orbit equidistributes towards vol Y 1 , where Y 1 is a union of components of Y ; (c) or x R F Y Y and its orbit equidistributes towards vol X .

  6.3. Invariant curves. By[START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF] Lem. 2.13], any action of a non-elementary group Γ on a projective surface X admits a maximal invariant curve D Γ , which can be easily detected from the action of Γ on H 2 pX, Zq since it corresponds to an invariant class. Bounds on the degrees of such invariant curves in terms of the action are given in[18, §3]. If in addition Γ contains a parabolic element, D Γ is the set of common components of the singular fibers of all elliptic fibrations associated to parabolic elements in Γ (see[20, §4.1]).

  X E is induced by a non-elementary subgroup G 0 of G. The standard linear action of G on C 2 (or more precisely on a neighborhood of any 2-torsion point of A) commutes to η and induces a linear action on C 3

	via the homomorphism
	(9.20)	φ › :	ˆa b c d	˙Þ ÝÑ	¨a2 ac ad `bc bc 2ab b 2 c 2 2cd d 2

3

by

(9.19)

φ : pu, vq Þ Ñ pu 2 , uv, v 2 q " px, y, zq, whose image is the quadratic cone txz ´y2 " 0u Ă C 3 . The level-2 congruence subgroup G of GL 2 pZq fixes each torsion point of A :" E ˆE of order ď 2, and Γ

Note that AutpXq is discrete unless X is a torus, see[START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF] Prop 3.16] so in most cases Γν " xSupppνqy.

Chung only assumes the moment condition (M) however it seems to us that a stronger assumption is needed for the control of the martingale differences.

Note that we use the invariance of µ here, not mere ν-stationarity.

We are slightly abusing here when the Fubini-Study metric depends on x, for instance when Px is not an isometry; however restricting to subsets of large positive measure the metric pPxq›pκ0qx is uniformly comparable to a fixed Fubini-Study metric.

Date: November 3, 2022.

Proof. The sets F and Y were already constructed in Theorem 10.1, whose proof also implies property (b). The classification of invariant measures (Theorem 6.4) and the stiffness property show that the only ν-stationary measure giving no mass to Y YF is vol X . Therefore the equidistribution property (c) follows from Breiman's ergodic theorem and the existence of a Margulis function associated to finite orbits and totally real surfaces (Theorems 4.1 and 4.5).

Remark 10.6. Note that if the maximal invariant totally real surface Y is known to be smooth (e.g. if Y " XpRq) then we may relax the finiteness assumption on ν to the moment condition (M `).

APPENDIX A. SYNCHRONIZING MEASURES ON PROJECTIVE SURFACES

Following [START_REF] Malicet | Random walks on HomeopS 1 q[END_REF], we say that ν is synchronizing if for every pair of points px, yq P M 2 , there is a measurable subset Ωpx, yq Ă Diff 1 pM q N such that ν N pΩpx, yqq " 1 and

distpf n ω pxq, f n ω pyqq " 0 for every ω P Ωpx, yq. This property is in a sense opposite to uniform expansion, and it is commonly satisfied for group actions on the circle (see [START_REF] Malicet | Random walks on HomeopS 1 q[END_REF][START_REF] Kleptsyn | Convergence of orbits in random dynamical systems on a circle[END_REF]). When (A.1) is replaced by

we say that ν is exponentially synchronizing. For example, if ν is a probability measure on PGL 3 pCq " AutpP 2 q that satisfies a moment condition and generates a Zariski dense subgroup, then ν is exponentially synchronizing.

Theorem A.1. Let X be a complex projective surface. Let ν be a probability measure on AutpXq that satisfies the moment condition (M). If ν is exponentially synchronizing, there is a finite index subgroup Γ 0 ν of Γ ν and a Γ 0 ν -equivariant birational morphism π : X Ñ X 0 , with X 0 " P 2 or a Hirzebruch surface, which conjugates Γ 0 ν to a subgroup of AutpX 0 q.

We expect that the same result holds for synchronizing actions as well.

Proof. Let us start with three preliminary observations. Firstly, Γ ν cannot act by isometries (for any distance on X); in particular, Γ ν is infinite. The same remark applies to exclude Γ νequivariant fibrations π : X Ñ B, with B a curve, for which Γ ν would act isometrically on the base. Secondly, if there is an invariant volume the dynamics cannot be synchronizing; so X cannot be a torus, a K3 surface, or an Enriques surface. Thirdly, by Pesin theory, if there exists a ν-stationary measure with a positive Lyapunov exponent, then the dynamics cannot be exponentially synchronizing; so, by Corollary 5.4, Γ ν is elementary. Denote by Γ ν the induced subgroup of GLpH ˚pX, Zqq, and assume temporarily that this groups is infinite. By the classification of elementary groups of automorphisms (see [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF]Thm 3.2] or [19, §10]), two cases may occur :

(1) either Γ ν contains a finite index cyclic subgroup generated by a loxodromic map f ˚, for some f P Γ ν ;

(2) or Γ ν contains a finite index, infinite, unipotent group that preserves the class rF s of a Γ ν -invariant genus 1 fibration π : X Ñ B.

In case (1), we can furthermore conclude that Γ ν is virtually cyclic, since otherwise X would be a torus (see Proposition 3.16 as well as the proof of Theorem 10.3 in [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]), which would contradict the second preliminary remark. Consider the random walk induced by ν on the cyclic subgroup f Z Ă Γ ν . If it is recurrent, then there are arbitrarily large times n i such that f n i ω is bounded in AutpXq, and this prohibits synchronization. If it is not recurrent, then the maximal entropy measure of f lifts to a Γ ν -invariant measure with a positive exponent, and this contradicts the third preliminary remark. So, case (1) is excluded. In case (2), the action of Γ ν on the base of its invariant fibration can not factor through a finite group (this would contradict the first preliminary remark); then, X should be a torus, in contradiction with the second remark. Thus, we conclude that Γ ν is finite.

Denote by AutpXq ˝the connected component of the identity in AutpXq. Lieberman proved in [START_REF] Lieberman | Compactness of the Chow scheme: applications to automorphisms and deformations of Kähler manifolds[END_REF] that AutpXq ˝has finite index in the kernel of the natural representation AutpXq Ñ GLpH ˚pX ; Rqq; thus, Γ ν :" Γ ν X AutpXq ˝is an infinite, finite index subgroup of Γ ν , and dim AutpXq ˝ě 1. Now, we refer to the classification of projective surfaces for which AutpXq is non-trivial. If the Kodaira dimension kodpXq is positive, this dimension must be 1 and the Kodaira-Iitaka fibration gives an AutpXq-equivariant fibration such that the action of AutpXq on the base factors through a finite group; thus, this case is excluded by the first preliminary remark. If kodpXq " 0, X should be a torus, in contradiction with the second preliminary remark. Thus kodpXq " ´8. If X is a ruled surface over a curve B of positive genus, then AutpXq acts by isometries on the base of the fibration, in contradiction with the first preliminary remark. Therefore X is a rational surface. The group AutpXq 0 fixes each rational curve of negative self-intersection. Thus, up to finite index, there exists a Γ 0 ν -equivariant birational morphism π : X Ñ X 0 onto a minimal model of X. Such a minimal rational surface is isomorphic to P 2 or a Hirzebruch surface.

APPENDIX B. RIGIDITY OF ZERO ENTROPY MEASURES

Here we complete the proof of Theorem 8.8 by establishing the following result of independent interest.

Theorem B.1. Let X be a torus or a K3 or Enriques surface, and ν be a probability measure on AutpXq such that Γ ν is non-elementary. Assume that µ is a Zariski diffuse ν-stationary measure such that h µ pX, νq " 0. Then µ is Γ ν -invariant and for every f P Γ ν , h µ pf q " 0.

Proof. As in Theorem 8.8, we may assume that µ is ergodic as a stationary measure, and its Γ ν -invariance was already established there. Pick f P Γ ν . Assume by way of contradiction that h µ pf q ą 0, in particular f must be loxodromic. If µ is ergodic for f , then the result follows rather immediately from the measure rigidity theorem 11.1 in [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]. Indeed in that theorem we consider an ergodic measure µ of positive entropy for f and study the group of automorphisms of X preserving µ, under the additional assumption that µ is supported on a real surface. We reduce the argument to the case of Γ " xf, gy for some g, and divide the proof into 3 cases: (1) either there is a Γ-invariant measurable line field, or (2) there is a Γ-invariant pair of measurable