Serge Cantat

Romain Dujardin

INVARIANT MEASURES FOR LARGE AUTOMORPHISM GROUPS OF PROJECTIVE SURFACES

We classify invariant measures for non-elementary groups of automorphisms, on any compact Kähler surface X, under the assumption that the group contains a so-called "parabolic automorphism". We also provide finiteness results for the number of invariant, ergodic, probability measures with a Zariski dense support. CONTENTS 1. Introduction 1 2. Two examples 4 3. The dynamics of Halphen twists 6 4. Proof of Theorem A: preliminaries, first steps, and corollaries 17 5. Proof of Proposition 4.15 26 6. Semi-analyticity of Σ and complements 35 7. Finitely many invariant measures: proof of Theorem C 42 8. Invariant analytic surfaces which are not real parts 45 9. Invariant surfaces with boundary 48 Appendix A. Abelian surfaces 51

INTRODUCTION

1.1. Non-elementary groups and parabolic automorphisms. Let Γ be a group of automorphisms of a compact Kähler surface X. We say that Γ is non-elementary if its image Γ ˚in GLpH 2 pX; Zqq, induced by its action on the cohomology of X, contains a non-abelian free group. We refer to [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF] for a description of such groups of automorphisms. In particular, it is shown in [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF] that the existence of a non-elementary subgroup in AutpXq implies that X is a projective surface.

By definition, an element g of AutpXq is a parabolic automorphism if }pg n q ˚} grows quadratically with the number n of iterates, where }¨} is any operator norm on EndpH 2 pX; Cqq. Any parabolic automorphism g preserves some genus 1 fibration, and a group Γ Ă AutpXq containing a parabolic automorphism is non-elementary if and only if it contains two parabolic automorphisms preserving distinct fibrations [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]. 1.2. Classification. In this article, we classify probability measures on X which are invariant by a non-elementary group containing a parabolic automorphisms. Our first theorem was already announced in [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]:

Theorem A. Let X be a compact Kähler 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 nonatomic and supported on a Γ-invariant curve D Ă X; (c) There is a Γ-invariant proper algebraic subset Z of X, and a Γ-invariant, totally real analytic surface Σ of XzZ such that (1) µpΣq " 1 and µpZq " 0;

(2) Σ has finitely many irreducible components;

(3) the singular locus of Σ is locally finite; and (4) µ is absolutely continuous with respect to the Lebesgue measure on Σ, and (5) its density (with respect to any real analytic area form on the regular part of Σ) 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.

Remark 1.1.

(1) Recall that an analytic surface Σ in an open U Ă X is totally real if for every smooth point x of Σ, the (real) tangent space T x Σ contains a basis of the complex tangent space T x X; equivalently T x Σ and its image j X pT x Σq by the complex structure satisfy T x Σ ' R j X pT x Σq " T x X. (2) Thus, each of the four cases (a), (b), (c), and (d) is characterized by a property of the support Supppµq: being finite, Zariski dense in a curve, totally real, or equal to X. (3) Given any non-elementary group Γ Ă AutpXq, there is a unique maximal Γ-invariant curve D Γ Ă X (see § 4.1 below). The invariant algebraic set Z is independent of µ and admits an explicit description (see Propositions 4.9 and 4.15). It is made of components of D Γ together with a residual finite set.

Corollary B. Let X be a compact Kähler surface. Let Γ be a non-elementary subgroup of AutpXq that contains a parabolic element and does not preserve any proper algebraic subset of X. If µ is a Γ-invariant and ergodic probability measure on X, then µ is (a) either a measure with real-analytic density on a compact, smooth, totally real, and real analytic surface Σ of X; (b) or a measure with real-analytic density on X.

In the totally real case (c) of Theorem A, it is natural to inquire about the structure of Σ on the whole surface X, including Z. Under a mild geometric condition (AC) we are indeed able to show that Σ admits a semi-analytic extension across Z: this is the contents of Theorem A' in Section 6. Recall that semi-analyticity means that Σ is defined locally by finitely many analytic inequalities (see §5.1). Since it requires some additional concepts, the condition (AC) will be described only in § 6.3: it concerns the action of Γ on the singular fibers of the elliptic fibration invariant by a parabolic element of Γ. This condition is satisfied in many interesting cases (for instance if D Γ is empty) and can be checked on concrete examples.

In Sections 8 and 9, we provide examples showing that the geometric conclusions of Theorems A and A' are, in a sense, optimal. More precisely it is shown that in case (c), -Σ is not necessarily contained in the real part of X, for some real structure on X, in other words, it is not necessarily contained in the fixed point set of an anti-holomorphic involution of X (see Corollary 8.2); -Σ can have a non-empty boundary (see §9.2).

1.3. Finitely many invariant measures. Theorem A is a key ingredient of the finiteness results for the number of periodic orbits in [START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF]. It also leads to the following alternative which is reminiscent of, but independent from, [START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF]Thm B].

Theorem C. Let X be a compact Kähler surface, and let Γ be a non-elementary subgroup of AutpXq containing a parabolic element. Then there are only finitely many ergodic Γ-invariant probability measures giving no mass to proper Zariski closed subsets, unless pX, Γq is a Kummer group.

We refer to [START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF] for the definition of Kummer groups; roughly speaking it means that the dynamics of Γ on X comes from the dynamics of a group of automorphisms on some torus C 2 {Λ. This result will be established in Section 7. We also show in § A.3 that a Kummer group can indeed admit infinitely many ergodic invariant measures of totally real type. Together with [10, Thm. C] (finitely many finite orbits) we thus obtain: Corollary D. Let X be a compact Kähler surface which is not a torus. Let Γ be a subgroup of AutpXq which contains a parabolic element and does not preserve any algebraic curve. Then there are at most finitely many Γ-invariant, ergodic probability measures on X.

1.4. Notes. A weak form of Theorem A is proven in [START_REF] Cantat | Sur la dynamique du groupe d'automorphismes des surfaces K3[END_REF] in the special case of K3 surfaces 1 . The results of [START_REF] Cantat | Sur la dynamique du groupe d'automorphismes des surfaces K3[END_REF] do not describe the support of the measure or the smoothness of its density, and are not sufficient to derive the global structure of Σ given by Corollary B, nor the finiteness result of Theorem C. In Section 3 we collect some preliminary results on genus 1 fibrations, their singular fibers, and the dynamics of automorphisms preserving such fibrations. We hope that this could prove 1 Also, one of the statements in [START_REF] Cantat | Sur la dynamique du groupe d'automorphismes des surfaces K3[END_REF] is slightly erroneous. In case (c), one first shows that µ gives mass to some germs of real analytic surfaces, and one has to glue these germs together to construct the surface Σ; to do so there is a monodromy problem which is overlooked in [START_REF] Cantat | Sur la dynamique du groupe d'automorphismes des surfaces K3[END_REF]. To say it differently, the results of [START_REF] Cantat | Sur la dynamique du groupe d'automorphismes des surfaces K3[END_REF] only imply that Σ is the analytic continuation of such a germ, which could a priori be dense in X. Overcoming this problem occupies a significant part of the present paper. useful beyond this paper (see also Duistermaat's monograph [START_REF] Duistermaat | Discrete integrable systems[END_REF] for a thorough treatment with a different focus). The core of the paper extends from Sections 4 to 7. A basic dichotomy is whether X is birationally equivalent to a torus, or not. The torus case relies on elementary tools from homogeneous dynamics, and the details are given in Appendix A. The proof of Theorem A occupies Sections 4 and 5. Theorems A' on the semi-analyticity of Σ and Theorem C are largely intertwined, and rest on a careful analysis of the action of the parabolic elements of Γ near the singular fibers of the associated elliptic fibrations. The details are given in Sections 6 and 7.

TWO EXAMPLES

2.1. K3 surfaces. Let X be any K3 surface. There is a holomorphic 2-form Ω X on X that does not vanish and satisfies ş X Ω X ^ΩX " 1; this form is unique up to multiplication by a complex number of modulus 1. Thus, the volume form vol X :" Ω X ^ΩX is AutpXq-invariant. If X comes with a real structure for which XpRq is non-empty, then XpRq is orientable and some multiple of Ω restricts to a positive area form on XpRq (see [START_REF] Silhol | Real algebraic surfaces[END_REF]§VIII.4] and §1 of [START_REF] Degtyarev | On the moduli space of real Enriques surfaces[END_REF]). This area form is multiplied by ˘1 by elements of AutpX R q; in particular, the measure induced by this form is invariant (see also Remark 2.3 below). We refer to [START_REF] Degtyarev | Real Enriques surfaces[END_REF][START_REF] Degtyarev | Topological properties of real algebraic varieties: Rokhlin's way[END_REF] for the topology of XpRq: it can be a sphere, the union of a sphere and a surface of genus 2, a torus, etc. Here are two explicit examples.

Example 2.1. (See [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF][START_REF] Cantat | Birational automorphism groups and the movable cone theorem for Calabi-Yau manifolds of Wehler type via universal Coxeter groups[END_REF]).-Take three copies of P 1 , with respective coordinates z i " rx i : y i s, i " 1, 2, 3. Let X Ă P 1 ˆP1 ˆP1 be a Wehler surface, i.e. a smooth surface of degree p2, 2, 2q; assume that X is very general in the family of such surfaces. In particular, X is smooth and it is a K3 surface. Fix an index k P t1, 2, 3u, let i ă j be the two indices such that ti, j, ku " t1, 2, 3u, and let π ij : X Ñ P 1 ˆP1 be the projection that forgets the k-th coordinate; this projection is a 2-to-1 cover, and we denote by σ k the involution that permutes the points in the fibers of π ij . Then, AutpXq is generated by the three involutions σ k and is non-elementary (see [START_REF] Cantat | Dynamique des automorphismes des surfaces K3[END_REF][START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF][START_REF] Wang | Rational points and canonical heights on K3-surfaces in P 1 ˆP1 ˆP1[END_REF]); the composition σ i ˝σj is a parabolic automorphism preserving the genus 1 fibration π k pz 1 , z 2 , z 3 q " z k ; and the composition σ 1 ˝σ2 ˝σ3 is a loxodromic automorphismwith topological entropy logp9`4 ? 5q ą 0. It is shown in [10, Thm A] that for a very general X, there is no AutpXq-invariant proper algebraic subset. If X is defined by a polynomial equation with real coefficients, then AutpXq preserves the real structure X R because the three involutions do. In particular, the real part XpRq is AutpXq-invariant.

For future reference let us note that the canonical invariant 2-form admits a simple explicit expression: consider affine coordinates x i P C corresponding to each of the three P 1 factors (with z i " rx i : 1s); then, X is defined by a polynomial equation P px 1 , x 2 , x 3 q " 0, and at every point in X, one of the partial derivatives of P does not vanish because X is smooth; then, up to some constant factor, (2.1)

Ω X " dx 1 ^dx 2 B x 3 P " dx 2 ^dx 3 B x 1 P " dx 3 ^dx 1 B x 2 P .
Example 2.2. (See [11, §3.2]).-Fix five lengths p 0 , 2 , . . . , 4 q P pR ˚q5 such that there is at least one pentagon P " pa 0 , . . . , a 4 q in R 2 , the sides of which satisfy pa i , a i`1 q " i (for i taken modulo 5); here, by a pentagon, we just mean an ordered set of five points a i in R 2 .

Assume that the family of such pentagons does not contain any flat pentagon (for instance this imposes 0 ` 1 ‰ 2 ` 3 ` 4 ). Consider the set of all such pentagons modulo affine positive isometries of R 2 ; thus, each pentagon can now be put in a normal position, with a 0 " p0, 0q and a 1 " p 0 , 0q. This set can be identified with a real algebraic surface XpRq that depends on p 0 , . . . , 4 q. There are five natural involutions acting algebraically on this surface: given one of the vertices a i of a pentagon P P XpRq, consider the two circles with centers a i´1 and a i`1 and respective radii i´1 and i , where indices are taken modulo 5; these circles intersect in two points a i and a 1 i ; thus we get an involution σ i of XpRq, mapping P to the pentagon σ i pPq with the same vertices except for a i that is replaced by a 1 i . Our hypotheses imply that XpRq is the real part of some real K3 surface X R and σ i P AutpX R q. Again, the composition σ i ˝σi`1 is a parabolic automorphism of X when the lengths are chosen generically.

Similar examples of large groups of automorphisms preserving a volume form on X (resp. a smooth measure on XpRq) can be constructed on some abelian surfaces and on most Enriques surfaces (see [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF] for instance).

Remark 2.3. If Σ is a totally real surface (of class C 1 , say) in an abelian or K3 surface X, which is invariant under a group Γ Ă AutpXq, then the canonical 2-form Ω X induces a Γinvariant measure on Σ. Indeed for every x P X, the tangent space T x X contains two C linearly independent vectors, thus Ω X | Σ induces a complex valued 2-form on Σ that does not vanish. Thus locally we can define an area form Ω Σ by Ω Σ,x " ξpxqΩ X,x where ξ is a function with values in the unit circle, and whenever Σ is orientable or not, this induces (by taking the associated density |Ω Σ | in the non-orientable case) the desired measure on Σ.

If X is an Enriques surface, the universal cover q : X 1 Ñ X is an étale 2-to-1 cover by a K3 surface. If Σ Ă X is a totally real surface, then its pre-image Σ 1 " q ´1pΣq is also totally real, and the automorphism of the covering q is an element of AutpX 1 ; Σ 1 q. Thus, applying the above construction in X 1 and pushing forward to X, we get an invariant measure on Σ as well.

Finally, if X is a blow-up of an abelian, K3, or Enriques surface, the same construction applies, except that the density of the associated volume form may vanish along the exceptional divisor of the blow-up.

2.2. Rational surfaces. The family of Coble surfaces (see [START_REF] Cantat | Rational surfaces with a large group of automorphisms[END_REF][START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]) and the examples described by Blanc in [START_REF] Blanc | On the inertia group of elliptic curves in the Cremona group of the plane[END_REF] give rational surfaces X such that AutpXq is non-elementary and contains parabolic elements (see [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF], §3.4, and Example 6.8 below). They are constructing by blowing up a finite number of points in P 2 : the 10 double points of some rational sextic S for Coble surfaces; a finite number of points on a cubic curve C for Blanc surfaces. The strict transform S 1 and C 1 of these curves are preserved by AutpXq. Denote by K X the canonical bundle of the surface.

(1) In the Coble case, there is a meromorphic section Ω of K b2 X that does not vanish and has a simple pole along S 1 .

(2) In Blanc's example, there is a meromorphic section Ω of K X that does not vanish and has a simple pole along C 1 .

In both cases, Ω induces a natural measure on X: for Blanc surfaces, it is given by the form Ω ^Ω; for Coble surfaces, it is given by Ω 1{2 ^Ω1{2 . In the Coble case, the total mass of this measure is finite, while in Blanc's example it is infinite. Moreover, if Γ Ă AutpXq is any subgroup generated by parabolic elements, then Γ preserves this measure. here, ZarpΓ ˚q is the Zariski closure in GLpH 2 pX; Rqq. For instance, pick finitely many parabolic automorphisms g i in AutpXq, and consider the group Γ generated by high powers g m i of the g i . If the g i do not preserve the same fibrations (see below), Γ is non-elementary; and if one chooses the g i correctly, Γ is thin. In case of Wehler surfaces, it suffices to take g 1 " σ 2 ˝σ3 and g 2 " σ 3 ˝σ1 and m " 3.

THE DYNAMICS OF HALPHEN TWISTS

3.1. Parabolic automorphisms and Halphen twists. An automorphism f of a compact Kähler surface X is said parabolic if

(3.1) ż X pf n q ˚κ ^κ -n 2
for some (hence any) Kähler form κ on X; equivalently, some power pf m q ˚of f ˚P GLpH 2 pX; Zqq is unipotent and the maximal size of its Jordan blocks is equal to 3; equivalently, f ˚acts on H 1,1 pX; Rq as a parabolic isometry with respect to the intersection form given by the cup product (see [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF]).

Theorem 3.1. Let f : X Ñ X be a parabolic automorphism of a compact Kähler surface.

(1) there exists a genus 1 fibration π : X Ñ B and an automorphism f B of the Riemann surface B such that π ˝f " f B ˝π; (2) if E is any (scheme theoretic) fiber of π, and F is a member of the linear system |E|, then F is a fiber of π;

(3) the foliation determined by the fibration π is the unique f -invariant complex analytic (smooth or singular) foliation on X; (4) f B has finite order, unless X is a compact torus.

The existence of the invariant fibration is proven in [START_REF] Gizatullin | Rational G-surfaces[END_REF] when X is a rational surface, and is easily obtained for other types of surfaces by the Riemann-Roch theorem (see [START_REF] Cantat | Dynamique des automorphismes des surfaces K3[END_REF]Proposition 1.4], and [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF] for a survey). The second assertion is not specific to invariant fibrations, but is good to be kept in mind; phrased differently, it says that the space of sections H 0 pX; OpEqq is 2-dimensional, and the divisor of zeroes divpsq 0 of any section s P H 0 pX; OpEqq is a fiber of π. The uniqueness of the invariant foliation and the last assertion are proven in [START_REF] Cantat | Symétries birationnelles des surfaces feuilletées[END_REF] (see Remark 3.10 below for the Kähler case).

When f B " id B we say that f is a Halphen twist. From the last item of the theorem, we see that if X is not a torus, for every parabolic f there exists k ě 1 such that f k is a Halphen twist. Beware that this terminology may differ from other references. Remark 3.2. Consider a fiber E of π, as in Assertion (2) of Theorem 3.1. Its class rEs P H 1,1 pX; Rq generates a ray R `rEs with the following property: if D is an effective divisor with rDs P R `rEs, then D is mapped to a point by π. Thus, R `rEs characterizes π as the unique fibration contracting these curves. In particular, an automrophism h of X permutes the fibers of π if and only if h ˚preserves R `rEs Ă H 1,1 pX; Rq.

3.2.

Complex and real analytic structures on fibers. Let π : X Ñ B be a genus 1 fibration on a compact Kähler surface X ( in this section, we do not assume that π is invariant by a parabolic automorphism). Our goal is to describe a foliation which is associated to π and the choice of a section of π (see [13, §2.1] for further details).

Denote by Critpπq Ă B the set of critical values of π, and by B ˝the complement of this finite set. On π ´1pB ˝q, π is a proper submersion and each fiber X w :" π ´1pwq is a curve of genus 1. Let U Ă B ˝be an open subset, endowed with -a holomorphic section σ : U Ñ X of π above U , -a continuous choice of basis of H 1 pX w ; Zq, for w P U .

If we declare that σpwq is the neutral element of X w , then X w becomes an elliptic curve for each w P U . There is, therefore, a unique holomorphic function τ from U to the upper half-plane H `Ă C such that -for every w P U , X w " C{Latpwq where

(3.2)
Latpwq " Z ' Zτ pwq » H 1 pX w ; Zq -the basis p1, τ pwqq of Latpwq corresponds to the chosen basis of H 1 pX w ; Zq.

Indeed, X U :" π ´1pU q is holomorphically equivalent to the quotient of U ˆC by the action of Z 2 defined by pp, qq ¨pw, zq " pw, z `p `qτ pwqq for pp, qq P Z 2 and pw, zq P U ˆC.

In the real-analytic category all one-dimensional complex tori are equivalent to R 2 {Z 2 as real Lie groups. Concretely, there is a unique isomorphism Ψ w : X w Ñ R 2 {Z 2 which maps the basis p1, τ pwqq of Latpwq to the canonical basis pp1, 0q, p0, 1qq of Z 2 ; in coordinates, if τ pwq " τ 1 pwq `iτ 2 pwq and z " x `iy, then

(3.3) Ψ w pzq " ˆx ´τ1 pwq τ 2 pwq y, 1 τ 2 pwq y ˙.
The real analytic diffeomorphism Ψ : π ´1pU q Ñ U ˆR2 {Z 2 defined by (3.4) Ψpw, zq " pw, Ψ w pzqq is the unique homeomorphism such that (1) π 1 ˝Ψ " π, where π 1 is the first projection; (2) Ψ maps the basis of H 1 pX w ; Zq to the canonical basis of Z 2 ; and (3) Ψ is an isomorphism of Lie groups in each fiber. In particular, Ψpw, σpwqq " pw, p0, 0qq.

In the following remarks, π 2 : U ˆR2 {Z 2 Ñ R 2 {Z 2 denotes the second projection.

Remark 3.3. For any pa, bq P R 2 {Z 2 , the holomorphic map w P U Þ Ñ a `bτ pwq P C{Latpwq determines a local section of π above U ; this section coincides with (3.5) Ψ ´1tpw, px, yqq ; px, yq " pa, bqu.

So, if we consider the real analytic foliation of U ˆR2 {Z 2 whose leaves are the fibers of π 2 , and if we pull-back this foliation by Ψ, we get a real analytic foliation F U of X U with holomorphic leaves, which will be referred to as the (local) Betti foliation. Now, consider the holomorphic map rks U : π ´1pU q Ñ π ´1pU q given by multiplication by some integer k ě 2 along the fibers of π; by definition, it fixes σpU q pointwise. Then, F U is invariant under the action of rks U . A leaf is pre-periodic if and only if it contains a torsion point of X w , for some and then for any w P U . The union of these preperiodic leaves is dense and each leaf of F U is a limit of such leaves.

Remark 3.4. Suppose that X is projective and σ is the restriction of an algebraic multisection of degree . This means that there is an irreducible curve C in X intersecting the general fiber of π in points such that the graph of σ is contained in C. Set k " `1. Then the multiplication map rks U extends as a rational transformation rks B : X X. Indeed, if X w is a general fiber and x is a point of X w , there is a unique point y such that p `1qx ´y is linearly equivalent to the divisor C X X w : by definition, rks B pxq " y.

If C is a section, i.e. " 1, F U extends globally to a foliation F of π ´1pB ˝q; we shall also refer to F as the (global) Betti foliation. The leaves of F corresponding to torsion points are, in fact, algebraic curves in X, since they correspond to the curves defined by r2s m`q B pxq " r2s m B pxq for some m ě 0 and q ě 1. On the other hand, the local projections π 2 ˝Ψ : π ´1pU q Ñ R 2 {Z 2 are not canonically defined; if γ is a loop in B ˝, with base point w 0 P U , then the analytic continuation of π 2 ˝Ψ along the loop is M pγq ˝π2 ˝Ψ, where M pγq P SL 2 pZq is given by the monodromy of the fibration (the determinant of M pγq is 1 because the orientation of the fibers is preserved). In other words, the monodromy of the fibration is induced by the holonomy of the Betti foliation. The section σ provides a fixed point σpw 0 q of the holonomy; the curves of pre-periodic points of r2s B correspond simultaneously to finite orbits of the holonomy group and to torsion points of the fiber X w 0 .

On the other hand, if C is a multisection of degree ě 2, the Betti foliation F U does not extend to B ˝: instead we obtain a web of degree at most , which is locally the superposition of the local Betti foliations associated to the choices of local sections whose graphs are contained in C. Remark 3.5. (see [START_REF] Cantat | The geometric Bogomolov conjecture[END_REF]). The form π 2 pdx ^dyq is a smooth closed form. Its pull back to X U is the local Betti form ω B " Ψ ˚π2 pdx ^dyq: (1) ω B is a closed semi-positive p1, 1q-form; (2) it vanishes along the leaves of F U (its kernel coïncides with T F U ); and (3) for w P B ˝, ω B|Xw is the unique translation invariant form of type p1, 1q such that ş Xw ω B, " 1. These properties characterize ω B, . If there is a global section, these forms patch together to define a global real analytic Betti form ω B on X B ˝(the monodromy group is contained in SL 2 pZq, so it preserves dx ^dy).

Note that the Betti form and Betti foliation depend on the choice of a section, but not on the choice of a basis of H 1 pX w 0 ; Zq.

3.3.

Singular fibers. Our goal in this section is to collect some facts concerning the geometry of a genus 1 fibration π : X Ñ B around one of its singular fibers. Furthermore, if f is a Halphen twist preserving π, we describe how its dynamical properties degenerate at a singular fiber, and how they are affected by the stabilization process, which reduces a singular fiber to a canonical model (see below). Of particular interest to us is the set of points w P B such that the orbits of f in X w are finite, or dense, or have a closure of dimension 1 (cf. Section 3.4). A first instance of stabilization is when π is not relatively minimal, that is when there is an exceptional curve of the first kind E contained in a fiber of π. There are finitely many such curves, so f permutes them, and some positive iterate f m fixes each of them. Thus, one can contract E in an f m -equivariant way, to end up with a birational morphism ε : X Ñ X 1 , a fibration π 1 : X 1 Ñ B such that π 1 ˝ε " π, and an automorphism f 1 of X 1 such that ε ˝f m " f 1 ˝ε. The dynamical properties of f 1 are the same as the ones of f : for example, the parameters w P B ˝such that each orbit of f in X w is dense coïncide with the parameters for which the orbits of f 1 in X 1 w satisfy the same property.

The local geometry of π around a critical value s P Critpπq was described by Kodaira. The reader is referred to [START_REF] Barth | of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF] for details, in particular Sections III.10, V.9, and V.10 there. From the above discussion, we may assume that X is relatively minimal. We fix s P Critpπq and further assume that X s is not a multiple fiber; the adaptation to the case of a multiple fiber will be described in § 3.3.3

Local sections.

A first observation is that when X s is not a multiple fiber there exists a local section of the fibration π around s. More precisely, for every component C of multiplicity 1 of π ´1psq, any small disk transverse to C is the graph of a section σ; and by Kodaira's classification, such a component always exists (see [1, §V.7]). Let us fix a small open disk V Ă B around s, such that V X Critpπq " tsu, together with such a local section σ : V Ñ X. Set X V " π ´1pV q and let X 7

V be the complement in X V of the irreducible components of X s that do not intersect σpV q (resp. of the singular point of X s if X s is irreducible); this set depends the chosen section. In other words, we keep from X s the smooth locus of the unique component intersecting σpV q; we shall denote by X 7 s Ă X 7 V this residual curve.

3.3.2. Type I b . The main example of singular fibers are those of type I b , with b P N ˚(type I 0 corresponds to the smooth case). For b " 1, X s is a rational curve with a unique normal crossing singularity; when b ě 2, X s is a cycle of b smooth rational curves of self-intersection ´2. So, X 7 s is biholomorphic to C ˆ" P 1 pCqzt0, 8u. Shrinking V if necessary, we can identify pV, sq with a disk pD R , 0q of radius R ă 1, and X 7

V with the quotient of D R ˆC by the family of lattices Latpwq " Z ' Zτ pwq given by (3.6) τ pwq " b 2iπ logpwq, for w P D R (note that logpwq is not well-defined but Latpwq is). For w " 0, the lattice Latpwq degenerates to Latp0q " Z Ă C. If γ is a loop making one positive turn around s, the monodromy M pγq maps the basis p1, τ pw 0 qq to p1, τ pw 0 q `bq. Let t : V Ñ C be a holomorphic function. The transformation g : V ˆC Ñ V ˆC defined by gpw, zq " pw, z `tpwqq induces, by taking the quotient, a holomorphic diffeomorphism of X 7

V . By [1, Prop. III.(8.5)], it extends to a diffeomorphism of X V that preserves π. Conversely, if f is a holomorphic diffeomorphism of X V that preserves each fiber of π, some positive iterate f m of f preserves each component of X s , and then f m maps σ to another section f m ˝σ intersecting X 7

s . Lifting to the universal cover, we see that there is a holomorphic function t : V Ñ C such that f is induced by pw, zq Þ Ñ pw, z `tpwqq; the function tpwq " f pσpwqq ´σpwq can be viewed as a section of the Jacobian fibration associated to π (see [1, §V.9] for details).

Consider the map

(3.7) pw, zq P D R ˆC Þ Ñ pw, vq :" pw, expp2iπzqq P D R ˆCˆ.
It is the quotient map for the action of Z Ă Latpwq on C by integral translations. If w ‰ 0, the vertical fiber twu ˆCˆi s mapped in X V to the elliptic curve C ˆ{w bZ » X w (because expp2iπτ pwqq " w b ). The fiber t0u ˆCˆi s mapped injectively onto the central fiber X 7 s of X 7

V .

Consider the Betti foliation F defined in π ´1pV ztsuq by the choice of the section σ. In D R ˆC, the leaves of F correspond to the curves pw, c `dτ pwqq, for pc, dq P R 2 . They are mapped in D R ˆCˆt o the curves γ c,d pwq " pw, expp2iπcqw bd q; here, |expp2iπcq| " 1 because c P R and w bd is multivalued as soon as bd R Z. Let us describe the local dynamics of F around X s . To simplify the exposition, we contract the components of X s that do not intersect the neutral section σpV q onto a point q; this gives a new surface X V . The central fiber of X V is irreducible and q is its unique singularity; when b ě 2, q is also a singular point of X V . By construction, X 7

V is biholomorphically equivalent to X V ztqu. When d " 0, the leaf defined by the curve γ c,0 extends to a local holomorphic section of π, given by pw, vq " pw, expp2iπcqq; the union of these curves is, locally, an F-invariant real 3-manifold which intersects the central fiber X 7 s » C ˆalong the unit circle tv P C ˆ; |v| " 1u. When d is rational, γ c,d pwq " pw, expp2iπcqw bd q extends to a local multisection of π; when σ is the restriction of a global section of π to the disk V Ă B, this local multisection γ c,d extends to an algebraic curve of X (a pre-periodic curve for r2s B , see Remarks 3.3 and 3.4). Finally, when d P RzQ, γ c,d is a transcendental and multivalued curve; in X V , the singularity q is the unique limit point of this curve on the central fiber.

Remark 3.6. In D R ˆCˆ, (3.8) αV " logp|v|q dw w ´logp|w|q dv v is a real analytic p1, 0q-form that vanishes along the curves γ c,d . Being invariant under the transformation pw, vq Þ Ñ pw, w b vq, it induces by taking the quotient a p1, 0q-form α V on π ´1pV ztsuq, the kernel of which coincides with the tangent space of the Betti foliation. To get the Betti form ω B defined by π and σ, one needs to multiply α V ^αV by a factor ϕpwq to ensure ş Xw ϕpwqα V ^αV " 1 for every w in V ztsu. The result is

ω B " i 2π ¨b 2plog |w|q 3 α V ^αV . (3.9)
3.3.3. Multiple fibers (see [1, §III.9 and V.10]). Let us assume in this paragraph that X s is a multiple fiber; it is necessarily of type mI b for some b ě 0. Let us do a local base change under the map p : ζ Þ Ñ ζ m " w; in other words, we consider the surface X 1 V given locally above V » D R by X 1 V " t pζ, xq P D R ˆXV ; πpxq " ζ m u, together with the projection π 1 : X 1 V Ñ D R defined by π 1 pζ, xq " ζ. Then, the map P : pζ, xq P X 1 V Þ Ñ x P X V satisfies π ˝P " p ˝π1 . The surface X 1

V may be singular, so we let X 2 V be the normalization of X 1 V and X pmq V be the minimal resolution of X 2 V ; there is a natural fibration π pmq : X pmq V Ñ D R and a natural map P pmq : X pmq V Ñ X V such that π ˝P pmq " p ˝πpmq . Now, it turns out that π pmq has no multiple fiber and that its central fiber (the unique possible singular fiber above V ) is of type I b .

If f is a holomorphic diffeomorphism of X V such that π ˝f " π, then f can be lifted to a holomorphic diffeomorphism f pmq of X pmq V such that P pmq ˝f pmq " f . First, one lifts f to X 1 V by pw, zq Þ Ñ pw, f pzqq and then to the normalization and its minimal resolution. Conversely, one recovers X V by taking the quotient of X pmq by the action of a finite group Z{mZ that commutes to f pmq . Thus, to study the local dynamics around multiple fibers, one only needs to study the case of fibers of type I b (including smooth fibers), and take a quotient by such a finite group.

3.3.4. Unstable fibers (see [1, §III.10 and V.10]). Let us now assume that X s is not multiple and not of type I b ; it is an unstable fiber of type II, III, IV , or I b , II ˚, III ˚, IV ˚. As in the previous paragraph, a local base change can be performed to end up with a local stable fibration; its central fiber will be smooth, except for the types I b , b ě 1 which lead to a central fiber of type I b . To do so, one first blows up the central fiber to ensure that its singularities are nodes, which gives rise to a new surface X V ; then, one does a base change to construct a new surface Y V (as above with X pmq ); a priori, the induced fibration on Y V is not relatively minimal anymore, so one contracts curves in the central fiber to construct a surface Y V with a relatively minimal fibration. Finally, X V can be recovered from Y V by taking a finite quotient, however only up to bimeromorphic equivalence (see [1, §III.10 and V.10]).

After taking some positive iterate f m , so that f m fixes each irreducible component of the singular fiber, the holomorphic diffeomorphism can be lifted to a holomorphic diffeomorphism of Y V ; indeed, f m induces a meromorphic map of Y V , and this map is a local diffeomorphism by [START_REF] Barth | of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]Prop. III(8.5)]. Thus, to study the dynamical properties of f we can focus locally on regular fibrations and singular fibrations of type I b , and then take a quotient by a finite group. Moreover, this finite group acts by multiplication by a root of unity on the base V » D R .

3.4. The dynamics of Halphen twists: twisting property. We pursue the study of a Halphen twist f : X Ñ X and of its invariant fibration π : X Ñ B. Let U Ă B ˝be an open disk, endowed with a section σ : U Ñ X of π and a continuous choice of basis of H 1 pX w ; Zq, for w P U . As in Section 3.2, there is a holomorphic function τ : U Ñ H `such that the fibers X w can be idendified to C{Latpwq, where Latpwq " Z `Zτ pwq. Along each fiber X w " C{Latpwq, f can be expressed in the coordinate z P C as ξz `tpwq, where t : U Ñ C is holomorphic. Here, ξ is a root of unity (of order dividing 12) which is determined by the action of f on H 1 pX w ; Zq » Z 2 and, as such, is locally constant; if ξ ‰ 1, f has finite order on each fiber X w , w P U , so f is periodic, and this contradicts the parabolicity assumption. Thus, f pzq " z `tpwq along X w , and f acts by translation along each fiber X w , w P U , of π. Now, conjugating f locally by the diffeomorphism Ψ : π ´1pU q Ñ U ˆR2 {Z 2 introduced in Section 3.2, f becomes (3.10) f Ψ pw, px, yqq " pw, px, yq `T pwqq for some real analytic map T : U Ñ R 2 . The following lemma says that "tpwq varies independently from τ pwq".

Lemma 3.7. The analytic map w P U Þ Ñ T pwq P R 2 {Z 2 is not constant. This is shown in [START_REF] Cantat | Sur la dynamique du groupe d'automorphismes des surfaces K3[END_REF], proof of Proposition 2.2, but the proof makes use of a global multisection of π, which exists if and only if X is projective 2 . For completeness, we present an argument which is more straightforward and applies to all Kähler surfaces. Remark 3.8. If σ is replaced by another local section σ 1 , the diffeomorphism Ψ is replaced by Ψ 1 " Φ ˝Ψ, with Φpw, px, yqq " pw, px, yq `Spwqq for some analytic map S : U Ñ R 2 {Z 2 . Then, f Ψ is changed into Φ ˝fΨ ˝Φ´1 and T is unchanged. Likewise, if the basis of H 1 pX w ; Zq is changed, T is mapped to A ˝T for some A P GL 2 pZq. Thus, the property that T is locally constant does not depend on the choices we made. Moreover, T is constant on U if and only if f preserves the Betti foliation F U associated to σ; thus, if this property holds above U for some choice of section, then it holds above any disk U 1 Ă B ˝and for any choice of local section.

Proof of Lemma 3.7. We fix a Kähler form κ on X, and compute the norm of the tangent map }Df } with respect to κ. Assume that T is constant on U . Then at each point pw, px, yqq of U ˆR2 {Z 2 , the tangent map pDf Ψ q pw,px,yqq is the identity; this implies that }Df n } is uniformly bounded above U , independently of n. By Remark 3.8, this property propagates over B ˝: if K Ă B ˝is any compact subset, there is a constant CpKq such that }Df n x } ď CpKq for all n ě 0 and all x P π ´1pKq.

Let us now study the behavior of f and Df near a singular fiber X s of type I b , for some s P Critpπq. Let V be a small disk around s, endowed with a section σ : V Ñ X, as in § 3.3.2. We identify V to D R and X 7

V to the quotient of D R ˆC by the family of lattices Latpwq " Z `Zτ pwq, with τ pwq as in Equation (3.6). Then, we change f into a positive iterate to assume that it fixes each irreducible component of the central fiber X s . From Remark 3.8 we know that the local Betti foliation F defined above V ztsu is f -invariant. As shown in Section 3.3.2, the leaves of F which extend to local sections of X 7

V foliate a unique local 3-manifold (that intersects X 7 s on a circle). Since f is a diffeomorphism preserving F and π, it preserves this 3-manifold. In the local coordinates pw, vq P D R ˆCˆi ntroduced in Section 3.3.2, this means that f acts by pw, vq Þ Ñ pw, hpwqvq for some holomorphic map w P D R Ñ hpwq P C ˆtaking its values in the unit circle. This shows that h is a constant of modulus 1, and that the iterates of f are locally contained, above V , in a compact group of local diffeomorphisms. Thus, }Df n } is also uniformly bounded around every singular fiber of type I b .

As explained in §3.3.4, we can contract curves and perform a stable reduction to get an automorphism g of a compact Kähler surface Y preserving a fibration π 1 : Y Ñ B 1 , a finite map η : B 1 Ñ B, and a meromorphic, dominant map ε : Y X such that π ˝ε " η ˝π1 and 2 The proof in [START_REF] Cantat | Sur la dynamique du groupe d'automorphismes des surfaces K3[END_REF] is written for K3 surfaces but extends to other projective surfaces ε ˝g " f ˝ε. The fact that T is constant on some open set is an intrinsic property, so it also holds for g. Since all singular fibers of π 1 are now of type I b , the previous argument shows that }Dg n } is uniformly bounded on Y ; in particular, }pg n q ˚}H 2 pY ;Rq is uniformly bounded. Let us show that this contradicts the parabolic behavior of f . Write ε as a composition ϕ ˝ψ´1 where ϕ : Z Ñ X is regular and ψ : Z Ñ Y is a bimeromorphic morphism. Set h " ψ ´1 ˝g ˝ψ, which is a bimeromorphic map of Z. By [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF]Prop. 1.15], }ph n q ˚}H 1,1 pZ;Rq is bounded. Then, (3.11)

ż Z pph n q › ϕ › κq ^pϕ › κq " ż Z ϕ › ppf n q › κ ^κq " degpϕq ż X pf n q › κ ^κ.
The left hand side is bounded because }ph n q ˚}H 1,1 pZ;Rq is bounded. This contradicts the parabolic behavior of f and concludes the proof.

Lemma 3.9. The differential DT w : T w U Ñ R 2 of T has rank 2 everywhere, except for finitely many points w j P U at which DT w j " 0. The analytic map T : U Ñ R 2 is an open mapping.

If DT w " 0, we shall say that f does not twist along the fiber X w ; this notion is intrinsic: it does not depend on the choice of the local diffeomorphism Ψ (the argument is the same as that of Remark 3.8). We define the set of non-twisting points by (3.12) NT f " tw P B ; DT w " 0u.

Proof of Lemma 3.9. Since f is holomorphic, its differential commutes with the complex structure j X of X. This implies that DT w ˝j " jpwq˝DT w where j is the complex structure on U Ă B and jpwq is a complex structure on R 2 {Z 2 that depends on w (it is the conjugate of the complex structure of C{Latpwq by the linear map DΨ w ). As a consequence, the rank of DT w is equal to 2 or 0. Assume that the real analytic set tw P U ; DT w " 0u contains some connected real analytic curve C Ă U . Along C, T is a constant T 0 ; then, the image f Ψ ptpw, p0, 0qq ; w P U uq of the zero section intersects the horizontal disk tpw, px, yqq ; px, yq " T 0 u on a real analytic curve. Now, let us come back to the complex surface X. The image f pσpU qq of the zero section and, by Remark 3.3, the set Ψ ´1tpw, px, yqq ; px, yq " T 0 u are two connected complex analytic curves. Since they intersect along a non-discrete subset, they coincide; this implies that T is constant, in contradiction with Lemma 3.7. Thus, the set of points at which DT w has rank ă 2 is locally finite. The last assertion follows easily from the first.

Remark 3.10. Lemma 3.9 shows that every f -invariant holomorphic foliation G on X is given by the invariant fibration π. Indeed, if T pwq P Q 2 {Z 2 and DT w ‰ 0, then a positive iterate f m fixes X w pointwise and, at every point x P X w , KerpDπ x q Ă T x X is the unique line invariant by the tangent map Df m x ; thus G must be everywhere tangent to X w and X w must be a leaf of G. Since T is open, T ´1pQ 2 {Z 2 q is dense in U , so G coincides with the fibration above U , hence everywhere on X. This proves Assertion (3) of Theorem 3.1 for compact Kähler surfaces ( [START_REF] Cantat | Symétries birationnelles des surfaces feuilletées[END_REF] considered only projective surfaces). Lemma 3.9 does not say that NT f is finite: it could cluster at a critical value of π. To exclude this possibility, let us first reformulate the twisting property. Consider an open set U Ă B endowed with a section σ and the induced local Betti foliation F. Its image f ˚pF q is generically transverse to F; these foliations are tangent at x P X U if, and only if DT πpxq " 0 if, and only if they are tangent along the fiber X πpxq . So, π ´1pNT f q " TangpF, f ˚pF qq. Lemma 3.11. Let s P Critpπq be the projection of a fiber of type I b , b ě 1. Let V be an open disk containing s, with a section σ : V Ñ X of π; let F be the Betti foliation determined by σ above V ztsu. If V is small enough, f ˚pF q is everywhere transverse to F above V ztsu.

Proof. We make use of Remark 3.6, and work in D R ˆCˆ. The automorphism f is given by f pw, vq " pw, hpwqvq for some holomorphic function hpwq " expp2iπtpwqq that does not vanish. In these coordinates, the neutral section σ is given by σpwq " pw, 1q, and its image under f is the curve f ˝σ : w Þ Ñ pw, expp2iπtpwqqq. The points w above which F is tangent to f ˚F are the points where the image of f ˝σ is tangent to F; they are determined by the equality αV ppf ˝σq 1 pwqq " 0, with αV as in Equation (3.8). This leads to the constraint (3.13) ´i logp|w|qwt 1 pwq " Imptpwqq;

writing tpwq " αw k `h.o.t., for some k in N, we get ´iαk logp|w|qw k » Impαw k q. As a consequence there is no solution close to the origin, and the proof is complete.

According to Section 3.3, the local dynamics of f near an unstable or multiple fiber is covered by the dynamics of a parabolic automorphism near a curve of type I b . So, the next result is a corollary of the previous lemmas. (Note that the existence of a section in Assertion (2) implies that X is projective and that π does not have multiple fibers 3 .) Proposition 3.12. Let f be a parabolic automorphism of a compact Kähler surface X, acting trivially on the base of its invariant fibration π : X Ñ B.

(1) The set of fibers π ´1pwq along which f does not twist is finite, i.e. NT f is finite.

(2) If π admits a global section σ : B Ñ X and F is the associated Betti foliation on π ´1pB ˝q, then TangpF, f ˚pF qq is a finite union of fibers.

3.5.

The dynamics of Halphen twists: orbit closure. We keep the notation from Section 3.4: we let U Ă B ˝be a disk, on which a continuous choice of basis for H 1 pX w ; Zq and a section of π have been chosen, so that H 1 pX w ; Zq » Z 2 and X w » R 2 {Z 2 .

A proper and closed subgroup of X w is finite or 1-dimensional.

A closed 1-dimensional subgroup L Ă X w is characterized by two data: a slope pp, qq, given by a primitive vector in Z 2 , and the number k ě 1 of connected components of L. The connected component of the identity

L 0 Ă L is the kernel of the homomorphism X w » R 2 {Z 2 Ñ R{Z defined by (3.14) px, yq Þ Ñ qx ´py,
and L is the preimage of the unique cyclic subgroup of order k in R{Z. Equivalently, L is the kernel of px, yq Þ Ñ kpqx ´pyq. We denote this subgroup by L w pk, pp, qqq. The integer k is intrinsically defined, but the slope depends on the basis of H 1 pX w ; Zq.

3 Indeed, if D Ă X is the graph of a section and F is a fiber, then F ¨D " 1, so F can not be multiple; and if F is reducible, D intersects F along a component of multiplicity 1. Moreover, if a P Z`is large enough, then aF `D is big and nef. Notation 3.13. For z P X, we denote by L f pzq the closure of the orbit of z, and by L 0 f pzq the connected component of z in L f pzq. For w in B, we denote by f w the restriction of f to the fiber X w . If πpzq R Critpπq, f πpzq is a translation in X πpzq ; thus L f pzq is either finite, or a translate of a 1-dimensional subgroup of X πpzq , or equal to X πpzq . By definition a translate of a connected, closed, 1-dimensional subgroup of X πpzq will be called a circle.

Define

TorpU q " tw P U ; f w : X w Ñ X w is a periodic translationu (3.15) " tw P U ; tpwq has finite order in X w " C{Latpwqu

" tw P U ; T pwq P Q 2 {Z 2 u.
This set is intrinsically defined (it does not depend on the section or on the basis of H 1 pX w ; Zq). By Lemma 3.9, TorpB ˝q is a countable and dense subset of B ˝. A point w belongs to TorpB ˝q if and only if the orbit of every z P X w is a finite subset of X w .

3.5.3. The next level of complexity is when tpwq belongs to a 1-dimensional subgroup of X w . Write f Ψ pw, px, yqq " pw, px, yq `T pwqq. For pα, βq in Q 2 {Z 2 and pp, qq P Z 2 ztp0, 0qu, we set R α,β p,q pU q " tw P U ; T pwq P pα, βq `R ¨pp, qqu. (3.16) This set depends only on the slope pp, qq and on qα ´pβ P R{Z, but not really on pα, βq itself. Using holomorphic coordinates, this real analytic curve R α,β p,q pU q is alternatively expressed as R α,β p,q pU q " tw ; the complex numbers tpwq ´σpα,βq pwq and p `qτ pwq are R-collinearu, where σ pα,βq is the local holomorphic section of π defined by (3.17) σ pα,βq pwq " Ψ ´1pw, pα, βqq " α `βτ pwq mod Latpwq;

here, as in Equation (3.4) , Ψ w maps the basis p1, τ pwqq of Latpwq Ă C onto the basis pp1, 0q, p0, 1qq of Z 2 Ă R 2 . This equation of R α,β p,q can be written locally as

(3.18) Im ˆtpwq ´σpα,βq pwq p `qτ pwq ˙" 0.
This curve may have singularities, as does the curve defined by Impw k q " 0 in the unit D when k ě 2; this happens precisely when w P NT f . Let R k p,q pU q Ă U be the closure of the set of points w for which L f pzq has k connected components of slope pp, qq for all z P X w ; it is the (finite) union of the R α,β p,q pU q, for all pα, βq in Q 2 {Z 2 such that qα ´pβ has order k in R{Z. Thus, R k p,q pU q is an analytic curve in U . For w P R k p,q pU qz TorpU q and z P X w , L f pzq is a translate of L w pk, pp, qqq. When performing an analytic continuation of R k p,q pU q around a critical value of π, the continuation may hit U again along a component of R k p 1 ,q 1 pU q for some new slope pp 1 , q 1 q; the vector pp 1 , q 1 q is in the orbit of pp, qq under the action of the monodromy group of the fibration. Since the orbit of pp, qq is typically infinite, the analytic continuation could a priori intersect U on infinitely many distinct R k p 1 ,q 1 pU q. Finally, for each integer k ě 1, we set

(3.19) R k pU q " ď pp,qqPZ 2 primitive
R k p,q pU q, and RpU q "

ď kě1 R k pU q.
These sets are intrinsically defined. Intuitively R k pU q should be thought as the set of w P U such that for z P X w , L f pzq is a union of k circles; formally, this is not a correct characterization of R k pU q because R k pU q contains points of TorpU q.

3.5.4. Summarizing this discussion, and keeping the same notation, we obtain:

Lemma 3.14. Let f be a parabolic automorphism of a compact Kähler surface acting trivially on the base of its invariant fibration π :

X Ñ B. Let U Ă B ˝be an open disk.
(1) The set TorpB ˝q is a dense and countable subset of B, and w P TorpB ˝q if and only if the translation f w is periodic, if and only if L f pzq is finite for every z P X w .

(2) For each slope pp, qq, the set R k p,q pU q is either empty, or a (possibly singular) real analytic curve; the set R k pU q is the union of these curves. Moreover, w P RpB ˝qz TorpB ˝q if and only if L 0 f pzq is a circle for each z in X w if and only if the closure of each orbit of f w is a union of circles embedded in X w ;

(3) If w P B ˝z RpB ˝q, then each orbit of f w is dense in X w .

In the second case, every f w -invariant and ergodic probability measure is the Lebesgue measure on L f pzq for some z P X w ; in the third case, the only f -invariant probability measure on X w is the Lebesgue measure.

3.6. Additional notes on the curves R α,β p,q pU q. Fix a finite set of real analytic arcs (slits) γ j Ă B intersecting transversally, and containing the critical values of π, such that the complement of Ť γ j in B is topologically a disk. Denote this disk by U . Then, choose a section σ of π and a continuous family of basis for H 1 pX w ; Zq above U . Each R α,β p,q pU q is a real analytic curve in U ; but when crossing an arc γ j , the analytic continuation of R α,β p,q pU q must be glued to another R α 1 ,β 1 p 1 ,q 1 pU q. In §6.2, we will have to understand the local structure of these curves near critical values of π and, as before, the important case is that of singular fibers of type I b . Here, we content ourselves with a few simple remarks, which help understand some of the subtleties of the problem of the semi-analytic extension of the curves σ g in Section 6.

Choose a local coordinate w as in § 3.3.2 and, to fix the ideas, suppose pα, βq " p0, 0q. From Equations (3.18) and (3.6), R 0,0 p,q is determined by the equation

(3.20) Im ˜tpwq p `q b 2iπ logpwq ¸" 0,
where logpwq is viewed as a multivalued function. The multivaluedness of the logarithm takes care of the monodromy in the sense that winding around the origin changes logpwq into logpwq` 2iπ, which in this equation corresponds to the monodromy action pp, qq Þ Ñ pp `qb, qq. The function tpwq is a well-defined local holomorphic function, which may or may not vanish at the origin (see §3.3.2). If q " 0, Equation (3.20) reduces to Imptpwqq " 0 which is an analytic subset of the disk (including the origin if Imptp0qq " 0). Now, we focus on the case q ‰ 0, and we assume that tpwq " t 0 w k . Write w " e ´s, where s belongs to some right half plane, and write t 0 " e ks 0 . Then, Equation (3.20) becomes (3.21) Im

˜e´kps´s 0 q p `q b 2iπ p´sq ¸" 0.
Writing s " x`iy and s 0 " x 0 `iy 0 , and making a few elementary manipulations, this equation reduces to (3.22) x " ˆy `2πp bq ˙tan kpy ´y0 q, which after a vertical translation gives x " py ´y1 q tanpkyq for some y 1 P R. There are two different regimes:

(1) if k " 0 the curve is a line, which descends to a logarithmic spiral in the w-coordinate;

(2) if k ą 0, one distinguishes two cases, depending on whether y 1 is of the form 1 k p π 2 `jπq, j P Z, or not (see Figure 1). In both cases, these curves have infinitely many branches asymptotic to the horizontal lines y " 1 k p π 2 `jπq, j P Z, as x Ñ `8. In the w-plane, these branches have well-defined tangent directions at the origin, so they extend as C 1 curves at 0, but they are not semi-analytic. FIGURE 1. The curves x " `y `π 4 ˘tan y (left) and x " `y `π 2 ˘tan y (right).

PROOF OF THEOREM A: PRELIMINARIES, FIRST STEPS, AND COROLLARIES

Let X be a smooth, compact, Kähler surface. Fix a subgroup Γ of AutpXq such that:

(i) Γ is non-elementary, (ii) Γ contains a parabolic element.
Then, as shown in [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF], X is a smooth complex projective surface. Our goal in this section is to prove Theorem A under the stronger assumption:

(ii') Γ contains a Halphen twist g (see § 3.1); equivalently Γ contains a parabolic automorphism acting with finite order on the base of its invariant fibration.

As explained in Theorem 3.1, this hypothesis automatically follows from (ii) when X is not an Abelian surface. For the easier case of Abelian surfaces, a direct proof of Theorem A is given in the Appendix.

Notation 4.1. If h is a Halphen twist, we denote by π h : X Ñ B h its invariant fibration, and by Singpπ h q the union of its singular and multiple fibers. For U Ă B h (resp. w P B h ), we set X h U " π ´1 h pU q (resp. X h w " π ´1 h pwq). Similarly, we make use of the notation R h pU q and Tor h pU q, with the index h. We denote by HalpΓq the set of Halphen twists h P Γ preserving each irreducible component of each fiber of π h . This set is invariant under conjugation: if f P Γ and h P HalpΓq, then f ´1 ˝h ˝f is an element of HalpΓq and its invariant fibration is given by π g ˝f .

Remark 4.2. The foliation defined by π h is uniquely determined by h, but π h itself is not canonically defined: post-composition by an automorphism of B h would give another projection defining the same fibration. Thus, the notation means that a projection π h was chosen for every fibration invariant by an element h P HalpΓq, the choice being the same for two twists preserving the same fibration. Then π h ˝f equals π f ´1˝h˝f up to post composition by an automorphism of the base. 4.1. Invariant curves, Tang and STang. According to [START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]Lemma 2.13] and [10, Section 3], there is a unique reduced, effective, and Γ-invariant divisor D Γ in X such that:

(1) the Γ-periodic irreducible curves C Ă X are exactly the irreducible components C i of D Γ ;

(2) the intersection form is negative definite on (4.1)

V pD Γ q :" VectprC i s, i " 1, . . . , kq Ă H 2 pX; Rq.

Remark 4.3. If g P HalpΓq, the divisor D Γ is made of irreducible components of fibers of π g , but D Γ does not contain any complete fiber: indeed the intersection form is negative definite on V pD Γ q Ă H 2 pX; Rq, while the self-intersection of a fiber is 0.

For pg, hq P HalpΓq 2 , we let Tangpπ g , π h q be the set of points x P X such that pdπ g dπ h qpxq " 0, and define

(4.2) Tang Γ " č pg,hqPHalpΓq 2 Tangpπ g , π h q
In plain words, x R Tang Γ if one can find g and h in HalpΓq such that π g and π h are transverse projections in a neighborhood of x (this is compatible with our convention for π g , see Remark 4.2). Note that by definition if F is a multiple component of a fiber of π g then dπ g ^dπ h " 0 along F for all h P HalpΓq (see §5.2.2 for more on this). Let Tang 1 Γ (resp. Tang 0 Γ ) be the union of the 1-dimensional (resp. 0-dimensional) components of Tang Γ . We also put

(4.3) STangpπ g , π h q " Tangpπ g , π h q Y Singpπ g q Y Singpπ h q and (4.4) STang Γ " č pg,hqPHalpΓq 2
STangpπ g , π h q, and define STang 1 Γ and STang 0 Γ similarly. Lemma 4.4. The reduced divisors given by D Γ , Tang 1 Γ and STang 1 Γ are all equal. The 0dimensional parts Tang 0 Γ and STang 0 Γ are finite Γ-invariant sets.

Proof. By definition, TangpΓq and STang Γ are Γ-invariant algebraic sets, and

STang 1 Γ con- tains Tang 1 Γ . So, Tang 1 Γ Ă STang 1 Γ Ă D Γ . Now, fix g P HalpΓq.
If C is a g-periodic irreducible curve, Lemma 3.14 entails that π g pCq must be a point. So, all fibrations π g must be tangent along any component of D Γ , and it follows that D Γ " Tang 1 Γ . The second assertion is straightforward.

Proposition 4.5 (See [START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF], Proposition 3.9). There is a normal projective surface X 0 and a birational morphism η : X Ñ X 0 such that (1) η contracts D Γ on a finite number of points;

(2) Γ induces a subgroup of AutpX 0 q, i.e. there is an injective homomorphism f P Γ Þ Ñ f 0 P AutpX 0 q such that η ˝f " f 0 ˝η for all f P Γ. Remark 4.7. Pick such a local analytic submanifold W Ă U with µpW q ą 0 and dim R pW q " d R pµq. By ergodicity, µ gives full mass to Ť gPΓ gpW q. If W X gpW q has empty relative interior in W and gpW q, then gpW q X W is contained in at most countably many analytic submanifolds of lower dimension, so µpgpW q X W q " 0.

Likewise, let d C pµq be the minimal dimension k P t0, 1, 2u such that µpZq ą 0 for some irreducible local complex analytic subset Z Ă X of (complex) dimension k. Lemma 4.8. Assume that Γ satisfies (i) and (ii'), and let µ be an ergodic Γ-invariant measure.

(1)

If d C pµq " 0, µ is supported on a finite orbit of Γ. (2) If d C pµq " 1, µ is supported on D Γ .
So, we see that d C pµq " 2 is equivalent to: µ gives no mass to algebraic sets (or more precisely to proper algebraic subsets).

Proof. The zero-dimensional case is left to the reader. So, suppose that µ has no atom and let Z Ă X be a local, 1-dimensional, complex analytic set such that µpZq ą 0. If Z is not contained in D Γ , one can find g P HalpΓq and a holomorphic disk ∆ Ă Z of positive measure which is transverse to π g . By Lemma 3.9, for any k ‰ , g k p∆q X g p∆q is a finite set; since µ is atomless and g-invariant, this implies µp Ť ně0 g n p∆qq " ř ně0 µp∆q " 8. This contradiction completes the proof. 4.3. The smooth case. For g P HalpΓq, the (marginal) measure µ g :" pπ g q ˚µ is a probability measure on B g . Let R g " RpB g q (cf. Equation (3.19)). Proposition 4.9. Let µ be an invariant and Γ-ergodic measure which gives no mass to proper algebraic subsets (i.e. d C pµq " 2q). Assume that there exists an element g P HalpΓq such that

(4.5) µ g pB g z R g q ą 0.
Then µ is absolutely continuous with respect to the Lebesgue measure on X and its support is equal to X. Moreover, the density of µ with respect to any real analytic volume form on X is a real analytic function in the complement of STang Γ .

The proof occupies § § 4. 2) holds also for h in place of g.

In Assertion [START_REF] Bierstone | Semianalytic and subanalytic sets[END_REF], the important part is that an f P xg, hy that permutes the fibers of π g is an element of g Z . Indeed, the following remark shows that the last three properties in Assertion (2) are always equivalent.

Remark 4.11. Consider an arbitrary pair pg, hq of Halphen twists, and let E be a scheme theoretic fiber of π g (that is, if w is a local coordinate near w 0 " π g pEq, the equation of E is π g pξq " w). If f P Γ maps E in a fiber E 1 of π h , then the class rf pEqs must be proportional to rE 1 s, because the self-intersection of f pEq is zero and in the vector space generated by the classes of the components of E 1 , all isotropic vectors are proportional to rE 1 s. Thus the classes of the fibers of π g ˝f and π h generate the same ray in H 1,1 pX; Rq; by Remark 3.2, π g ˝f ´1 and π h are equal, up to post composition by an isomorphism B h Ñ B g . In particular, if g and h preserve distinct fibrations, no fiber of π g is entirely contracted by π h . Lemma 4.12. If g and h are Halphen twists associated to distinct fibrations, then for large enough n, the pair pg n , h n q is special.

Proof. Consider the action of Γ on H 1,1 pX; Rq. It preserves the isotropic cone (4.6) tu P H 1,1 pX; Rq ; xu|uy " 0u,

where x¨|¨y is the intersection form. Projectively, this cone is a sphere S (which can be considered as the boundary of a hyperbolic space, see [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF][START_REF] Cantat | Random dynamics on real and complex projective surfaces[END_REF]). Now, g ˚is a parabolic transformation on S, fixing a unique point u g . Let us fix a small neighborhood U g of u g ; if x is a point of Sztu g u there is small neighborhood V pxq of x in S and a positive integer m g pV pxqq such that pg n q ˚pV pxqq Ă U g for every n P Z with |n| ě n g pV pxqq. A similar property is satisfied by h. The points u g and u h are determined by the classes of the fibers of the invariant fibrations π g and π h . We choose U g and U h small and disjoint, and n 0 such that for all n P Z with |n| ě n 0 (4.7) pg n q ˚pU h q Ă U g z tu g u and ph n q ˚pU g q Ă U h z tu h u .

In the following we fix such an n. Then, the first assertion follows from the ping-pong lemma (see [START_REF] De La Harpe | Topics in geometric group theory[END_REF], Chapter I).

Let us show that if f P xg n , h n y, then f ˚ug " u g if and only if f P xgy. We assume that f ˚ug " u g , and we want to show that f is an iterate of g. Write f as a word in g n and h n : f " g nk 1 ˝hn 1 ˝¨¨¨˝h n s , the k i and i being non-zero integers, except possibly for k 1 and s which may vanish. The proof is by induction on |f | " |ti ;

k i ‰ 0u| `|ti ; i ‰ 0u|. The result is obvious when |f | " 0 or 1.
The point u g corresponds to an isotropic line in H 1,1 pX; Rq, and this line intersects H 2 pX; Zq on a discrete set ZrDs, for some integral class rDs on the boundary of the ample cone. Since f preserves H 2 pX; Zq and the ray R `rDs, it fixes rDs. Thus, f ˚cannot be loxodromic (see [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF]).

If f starts with h and ends with g (i.e. s k 1 ‰ 0), then f ˚maps U h strictly inside itself and pf ˚q´1 maps U g strictly inside itself. This implies that f ˚is loxodromic, a contradiction. Therefore f starts and ends with the same letter. If this letter were h, then u g would in fact be mapped into U h by f ; thus, f starts and ends by a power of g. Conjugating f by a power of g, we reduce its length without changing the property f ˚ug " u g . Thus, by induction, f is a power of g.

4.3.2.

Preliminaries: disintegration. If g is any Halphen twist, we may disintegrate µ with respect to π g : there is a measurable family of probability measures µ g,w on the fibers X g w such that g,w is another family of probability measures satisfying Equation (4.8), then µ 1 g,w " µ g,w for µ g -almost every w. Thus, the measures µ g,w are invariant under the action of g |X g w .

4.3.3. Proof of Proposition 4.9: special case. In this paragraph we assume that there exists h P HalpΓq such that the pair pg, hq is special in the sense of Definition 4.10, and let Γ 1 " xg, hy. Note that Γ 1 satisfies assumptions (i) and (ii') from p.17. Let µ be as in the statement of Proposition 4.9. Let us show that the conclusions of the proposition hold under the additional assumption that that µ is Γ 1 -ergodic (also in this first part of the proof, the analyticity of the density will only be established outside STangpπ g , π h q).

Let B g be a Borel subset of B g which is disjoint from R g and satisfies µ g pB g q ą 0. According to Lemma 3.14, the dynamics of g on X g w is uniquely ergodic for every w P B g . Thus, we get:

Step 1.-If w P B g , then g |X g w is uniquely ergodic and µ g,w is equal to the Haar measure λ g,w on the fiber X g w » C{Lat g pwq. Here, Lat g pwq " Z ' Zτ g pwq and λ g,w " pImpτ g pwqqq ´1idz dz, as in Section 3.

Step 2.-We have d C pµq " 2 and µ g pR g q " 0.

The first assertion follows directly from Lemma 4.8 and Lemma 4.4. For the second one, we argue by contradiction, assuming that there is an analytic arc γ Ă R g such that µ g pγq ą 0. Since d C pµq ą 1, we can shorten γ to ensure that it does not contain any critical value of π g . Set W γ " π ´1 g pγq. Then µpW γ q ą 0, so d R pµq ď 3. By ergodicity of µ,

Ť f PΓ 1 f pW γ q is a subset of full measure. Pick f P Γ 1 .
If it permutes the fibers of π g , then f P g Z because pg, hq is special, and thus f pW γ q " W γ ; thus µpf pW γ q X π ´1 g pB g qq " µpHq " 0 in that case. Now, suppose f does not permute the fibers of π g . Note that if W γ intersects an irreducible curve C Ă X on some non-empty open subset of C, then C must be a fiber of π g , because its projection in B g is locally contained in γ. Thus, if w P B g and f pW γ q X X g w contains a non-empty open subset of X g w then f ´1 maps X g w into a fiber of π g above γ, and this contradicts the fact that pg, hq is special. We deduce that f pW γ q X X g w is contained in a countable union of real analytic submanifolds of dimension 1 in X g w , for every w P B g . In particular if w P B g , λ g,w pf pW γ q X X g w q " 0, and we conclude that µpf pW γ q X π ´1 g pB g qq " 0 in that case too. Since µ `ŤfPΓ 1 f pW γ q ˘" 1 and simultaneously µpπ ´1 g pB g qq ą 0 we obtain a contradiction, which concludes the proof of the second step.

Step 3.-From Step 2 we can suppose µ g pB g q " 1. Let us show that µ h :" pπ h q ˚µ is absolutely continous with respect to the Lebesgue measure. In particular, h satisfies µ h pB hz R h q " 1 too.

Let ∆ Ă B h be a Borel set of Lebesgue measure 0. Remark 4.11 shows that if X g w is a smooth fiber of π g , then X g w is not contracted by π h , so π h|X g w : X g w Ñ B h is a finite ramified cover, and λ g,w ppπ h|X g w q ´1p∆qq " 0. This shows that (4.9) λ g,w pπ ´1 h p∆q X X g w q " 0 for µ g almost every point w P B g . Since µ g pB g q " 1 and µ g,w coincides with the Haar measure λ g,w for µ g almost every w P B g , Equation (4.8) implies µpπ ´1 h p∆qq " 0; thus µ h p∆q " 0, as required.

Step 4.-The support of µ is X.

Indeed, µ h being absolutely continuous with respect to the Lebesgue measure of B h , µ h pR h q " 0. Thus, symetrically, µ g is absolutely continuous with respect to the Lebesgue measure of B g , and from Equation (4.8) we deduce that µ is absolutely continuous with respect to the Lebesgue measure on X. If U is an open subset of B g , X g U intersects every smooth fiber of π h on a set of positive Haar-measure; thus µpX g U q " µ g pU q is positive, and we infer that the support of µ g is equal to B g . Since µ g,w is µ g -almost surely the Haar measure λ g,w , we conclude that the support of µ is equal to X.

Step 5.-The density is analytic outside STangpπ g , π h q.

Let x 0 be a point of XzSTangpπ g , π h q and V be a small neighborhood of x 0 such that π g and π h are everywhere transverse on V , -U g :" π g pV q and U h :" π h pV q are small disks in B g and B h , respectively, -π g (resp. π h ) is a proper submersion above U g (resp. U h ).

In a chart Ψ g mapping X g Ug to U g ˆR2 {Z 2 , Equation (4.8) implies that pΨ g q ˚pµ |X g Ug q is invariant under the action of all vertical translations; the same property holds with respect to π h . Coming back to X, these translations act analytically and locally transitively on V : for every y in V there is a pair of such translations such that their composition maps x 0 to y (V is not invariant under such translations). Following the proof of [22, Proposition 1], we deduce that µ has a real analytic density on V , with respect to the analytic structure of X. Indeed, embed V in R 4 and denote by vol a volume form on R 4 . In these coordinates, µ " ξ vol, for some nonnegative integrable function ξ. Changing x 0 P V if necessary, we suppose that x 0 is a Lebesgue density point for µ; this means that µpεKqvolpεKq ´1 converges to ξpx 0 q when ε Ñ 0, for every ellipsoid K centered at x 0 . If x 0 is mapped to y by a diffeomorphism ϕ y preserving µ, then the boundedness of the distortion of ϕ y shows that y is also a Lebesgue density point of µ, with density ξpyq " pJacpϕ y qpx 0 qq ´1ξpx 0 q. Now, choosing ϕ y as a composition of two translations, we can assume that y Þ Ñ Jacpϕ y qpx 0 q is an analytic function; thus, the density ξ is real-analytic in a neighborhood of x 0 . 4.3.4. Proof of Proposition 4.9: general case. Let µ and g be as in the statement of the proposition. Fix h P HalpΓq such that dπ g ^dπ h ı 0. Then by Lemma 4.12, we find n ě 1 such that pg n , h n q is special; set Γ 1 " xg n , h n y. Since R g n " R g , the assumption (4.5) holds with g n instead of g. However µ is not necessarily Γ 1 -ergodic, so the results of § 4.3.3 cannot be applied directly. To get around this difficulty, we use the Γ 1 -ergodic decomposition of µ (see [START_REF] Varadarajan | Groups of automorphisms of Borel spaces[END_REF]): µ " ş X β x dµpxq for an essentially unique, Γ-invariant, Borel map β : x Þ Ñ β x such that β x is µ-almost surely a Γ 1 -ergodic probability measure.

Set Ω j " tx P X ; d C pβ x q " ju, for j " 0, 1. By ergodicity, β x pD Γ 1 q " 1 for every x P Ω 1 . Since µpD Γ 1 q " 0, we deduce that µpΩ 1 q " 0. For x P Ω 0 , β x gives full mass to the union of the (fixed) countable set π ´1 g pTorpB g qq X π ´1 h pTorpB hqq and the Zariski closed set Singpπ g q Y Singpπ h q; since d C pµq " 2, we also get µpΩ 0 q " 0. Thus dim C pβ x q " 2 for µ-almost every x.

So, for x in a subset Ω Ă X with µpΩq " 1, the results of § 4.3.3 apply to β x . There are two possibilities:

-either β x pπ ´1 g pB g z R g qq ą 0, β x is absolutely continuous, and its support is X; -or β x pπ ´1 g pR g qq " 1. Denote by Ω ac (for "absolutely continuous") and by Ω si (for "singular") the set of points such that the first or second alternative holds, respectively. Both are Borel subsets and µpΩ ac YΩ si q " 1. Assumption (4.5) implies that µpΩ ac q ą 0. If µpΩ si q ą 0, then µpπ ´1 g pR g qq ą 0, and since µ is Γ-ergodic, we infer that µpΓ ¨π´1 g pR g qq " 1. But the Lebesgue measure of Γ ¨π´1 g pR g q is zero, so ş xPΩac β x pΓ ¨π´1 g pR g qq dµpxq " 0 and we deduce that µpΩ ac q " 0. This contradiction shows that µpΩ si q " 0 and µpΩ ac q " 1.

Finally, if x and y are in Ω ac , then β x and β y are Γ 1 -ergodic measures of full support, the densities of which are analytic on the complement of proper analytic sets. So β x " β y , and this implies that β x " µ almost surely. In particular, µ is Γ 1 -ergodic and satisfies the conclusions of § 4.3.3. Remark 4.13. This argument shows that if µ is a Γ-ergodic probability measure, µ gives no mass to algebraic subsets, and µ g pB g z R g q ą 0 for some g P HalpΓq, then µ g 1 pB g1 z R g 1 q " 1 for any g 1 P HalpΓq: this follows from the absolute continuity of µ.

To conclude the proof of Proposition 4.9, it remains to show that the density of µ is analytic outside STang Γ . For every x P XzSTang Γ there exists a pair pg 1 , h 1 q P HalpΓq 2 such that x R STangpπ g , π h q, and by Lemma 4.12 we can assume that this pair is special. By the previous remark, the results of § 4.3.3 apply to pg 1 , h 1 q. So, µ is smooth near x, and we are done. Proof. By Lemma 4.8, we may assume that d C pµq " 2, in particular µ g is atomless, and we seek a contradiction. Pick g P HalpΓq. By assumption there is a (local) real analytic curve W Ă X such that µpW q ą 0 and W is Zariski dense in X. Shrinking it, we can assume that W is an analytic path, transverse to the fibration π g , that intersects each fiber above π g pW q in a unique point. By Proposition 4.9, π g pW q is contained in R g . Let us disintegrate µ with respect to π g . Since µ g is atomless, Lemma 3.14 shows that, for µ g -almost every w, the measures which are invariant under g w are atomless; thus, the conditional measure µ g,w is almost surely atomless and µ g,w pW X X g w q " 0. This contradicts µpW q ą 0.

4.5. The totally real case. We are now reduced to the case where d R pµq ě 2 and µ g is supported on R g for every g P HalpΓq. The properties of µ are summarized in the following proposition, the proof of which will be given in Section 5.

Proposition 4.15. Let µ be an ergodic Γ-invariant measure. Assume that µ gives no mass to algebraic subsets and that µ g pR g q ą 0 for some g P HalpΓq. Then d R pµq " 2, and there exists a totally real analytic subset Σ of XzSTang Γ of pure dimension 2 such that:

(1) µpΣq " 1 and Supppµq " Σ;

(2) Σ has finitely many irreducible components;

(3) the singular locus of Σ is locally finite;

(4) on the regular part of Σ, µ has a real analytic density with respect to any real analytic area form;

Example 4.16. The main example is when the projective surface X is defined over R, and µ is a measure supported on XpRq giving no mass to algebraic subsets. This occurs for instance when Γ Ă AutpX R q preserves an area form on XpRq (see Section 2 Let us point out the following immediate consequence of Theorem A.

Corollary 4.18. If X is a real projective surface, XpRq is nonempty, Γ Ă AutpX R q, and µ is a Γ-invariant ergodic probability measure on XpRq, then (a) either µ is supported by a Γ-invariant proper real algebraic subset of X, (b) or there is a Γ-invariant proper real algebraic subset Z of X such that µ is supported by a union of connected components of XpRqzZpRq and µ has a real analytic density on that set.

In particular if Γ does not preserve any proper real algebraic subset of X, µ is given by a real analytic area form on XpRq, restricted to a Γ-invariant union of connected components of XpRq.

Let X be (a blow up of) an abelian, K3, or Enriques surface, and let vol X be the natural probability measure on X (see 2.1). Likewise, if Σ is a totally real submanifold in X, let vol Σ be the measure induced by the normalized 2-form Ω X (see Remark 2.3). By positivity of the density of µ and ergodicity, we obtain: Corollary 4.19. If in Theorem A, X is a blow-up of an abelian, K3, or Enriques surface, then in case (c) µ is a multiple of vol Σ , and in case (d) it coincides with vol X .

In case (c), this implies in particular that Σ has finite area. Similarly, case (d) does not appear in Blanc's example (see § 2.2). PROPOSITION 4.15 In this section, we prove Proposition 4.15. Thus, unless otherwise specified, we study ergodic invariant measures with d C pµq " 2 and µ g pγq ą 0 for some g P HalpΓq and some local, smooth, analytic arc γ Ă R g . The main step of the proof is to show that γ extends to an analytic curve σ in B g . From Proposition 4.9, we already know that the support of µ g is contained in the closure of R g ; however the analytic continuation of γ and the support of µ g still could a priori be a complicated subset in B g (see Remark 5.8). The main point is to exclude such a phenomenon.

PROOF OF

After some preliminaries in § § 5.1 and 5.2, the proof of Proposition 4.15 is carried out in § § 5.3 to 5.7.

5.1.

Vocabulary of analytic and semi-analytic geometry. Let us recall briefly a few basic facts from analytic and semi-analytic geometry. A semi-analytic set E in a (σ-compact) realanalytic manifold M is a subset such that every x P M admits an open neighborhood U in which E X U is defined by finitely many inequalities of the form f ě 0 or f ą 0, where f : U Ñ R is analytic. The class of semi-analytic sets is stable under many operations such as taking a finite union or intersection, the closure, the boundary, the connected components, or the preimage under an analytic map. However the image of a semi-analytic set by a proper analytic map needs not be semi-analytic: adding such projections, one obtains the class of subanalytic sets.

The main fact that we will need from subanalytic geometry is: any subanalytic set of dimension ď 1 is semi-analytic. In this way we will only have to deal with semi-analytic sets (see [START_REF] Bierstone | Semianalytic and subanalytic sets[END_REF][START_REF] Łojasiewicz | Sur la géométrie semi-et sous-analytique[END_REF] for more details on these facts).

A point x in E is regular if E is an analytic submanifold in the neighborhood of x, otherwise E is singular at x. The dimension of E is the maximal dimension of E at its regular points. We say that E is of pure dimension if its dimension is the same at all regular points.

A delicate point in semi-analytic and real-analytic geometry is that the notion of irreducible component is not well behaved (we will discuss only the analytic case). To get around this problem, the following notion was introduced by Cartan [START_REF] Cartan | Variétés analytiques réelles et variétés analytiques complexes[END_REF] and Whitney and Bruhat [START_REF] Whitney | Quelques propriétés fondamentales des ensembles analytiques-réels[END_REF]: a subset E of a real analytic manifold M is C-analytic (or global analytic) if it the common zero set of a finite number of analytic functions defined on the whole of M . Equivalently, there is a coherent analytic sheaf whose zero set is E (see [START_REF] Whitney | Quelques propriétés fondamentales des ensembles analytiques-réels[END_REF]Prop. 10]). This class is stable by union and intersection. Every C-analytic set E admits a unique locally finite decomposition E " Ť i E i into irreducible components; here, the irreducibility means that E i is not the union of two distinct C-analytic sets (beware that it might be reducible as an analytic set, see [32, §11.a]). If E Ă M is a smooth analytic submanifold, or if E is locally a finite union of smooth plaques of the same dimension, then E is C-analytic. Indeed, in this case it is easy to see that for every x P E there exists an open neighborhood U Q x and a finite family of analytic functions f i on U such that for every open subset V Ă U , the intersection of the zero sets of the f i coincides with E in V : this implies that E is the zero set of a coherent analytic sheaf, hence it is C-analytic. Another useful fact is that every 1-dimensional analytic set is C-analytic; more generally, if E is analytic, the canonical ideal sheaf of analytic functions vanishing on E is coherent outside a codimension 2 subset of E, (see [START_REF] Galbiati | Stratifications et ensemble de non-cohérence d'un espace analytique réel[END_REF]).

A semi-analytic subset Σ Ă X of pure dimension 2 is totally real if at every regular point x P Σ, T x Σ is a totally real subspace of T x X, that is, j x pT x Σq ' R T x Σ " T x X, where j x is the complex structure (multiplication by ? ´1).

5.2. Preliminaries and conventions.

5.2.1.

Choice of Halphen twists. Recall from Remark 4.13 that under the assumptions of Proposition 4.15, µ g pR g q " 1 for any g P HalpΓq. We fix a pair of elements pg, hq P HalpΓq 2 associated to different fibrations. This pair will be kept fixed until Subsection 5.7 . Recall from Notation 4.1, that we denote by X g w the fiber π ´1 g pwq, and similarly for h.

5.2.2.

A decomposition of the tangency locus. By definition, the tangency locus Tangpπ g , π h q is the locus where the map pπ g , π h q : X Ñ B g ˆBh is not a local diffeomorphism; Tangpπ g , π h q is a curve that contains all multiple components of fibers of π g and π h , as well as the curves along which the foliations determined by these fibrations are tangent. To be more precise, we split Tangpπ g , π h q into four parts, (5.1) Tang ff pπ g , π h q Y Tang ft pπ g , π h q Y Tang tf pπ g , π h q Y Tang tt pπ g , π h q which are defined as follows. An irreducible component C of Tangpπ g , π h q is -contained in Tang ff pπ g , π h q if and only if C is contained in a fiber of π g and in a fiber of π h (in that case, C is both g and h-invariant, and its self intersection is negative, see § 4.1); -contained in Tang ft pπ g , π h q if and only if C is a multiple component of a fiber of π g but is generically transverse to π h ; -contained in Tang tf pπ g , π h q if and only if C is a multiple component of a fiber of π h but is generically transverse to π g ; -contained in Tang tt pπ g , π h q if it is generically transverse to both fibrations.

The superscripts f, t stand for fiber and transverse. Lemma 5.1. If Tang tt pπ g , π h q is empty, the fibrations π g and π h are both isotrivial.

The isotriviality of π g means that the j-invariant of the fibers X g w is constant on B g . In this case, the discussion of § 3.3.2 shows that no fiber of π g is of type I b or I b , b ě 1, and that after a finite base change, π g becomes birationnally equivalent to a trivial fibration.

Proof. If Tang tt pπ g , π h q is empty, the foliation associated to π h is transverse to π g on π ´1 g pB g q (though a multiple fiber of π h may intersect every fiber of π g ). If β : r0, 1s Ñ B g is a smooth path, the holonomy of this foliation determines a holomorphic diffeomorphism hol β : X g βp0q Ñ X g βp1q . Thus, all fibers of π g are isomorphic, and likewise for π h .

Geometry of g-orbits.

Let us fix a Kähler metric on X, given by a Kähler form κ, as well as a Kähler form κ g on B g (hence also on B h , see § 5.2.1). Lengths, areas, and diameters will be computed with respect to these metrics. According to the Notation 3.13, a circle in an elliptic curve is a translate of a 1-dimensional, closed, and connected subgroup. Being invariant under translation, this notion is well defined on X g w , for every w P B g . If w P R k g and z P X g w , the closure of its g-orbit is a union of k circles; the circle containing z is denoted L 0 g pzq. In the next lemma we use the notation introduced in Section 3.5; the norm of a slope pp, qq is, by definition }pp, qq} " pp 2 `q2 q 1{2 . Since circles are homotopically non-trivial and long circles become asymptotically dense, we get: Lemma 5.2. Let U Ť B g be a disk, endowed with a continuous choice of basis for H 1 pX g w ; Zq and a local section of π g .

(1) There is a real number pU q such that the length of every circle of every fiber X g w , for w P U , is bounded from below by pU q.

(2) For every ε ą 0, there is a real number D ą 0 such that for all pp, qq with }pp, qq} ą D, and all w P U , every circle of slope pp, qq is ε-dense in X g w . In particular, for every ε ą 0, there is a real number D ą 0 such that if }pp, qq} ą D, and w P R k p,q pU qz TorpU q for some k then the circle L 0 g pzq is ε-dense in X g w for every z P X g w .

(3) For every ε ą 0, there is an integer k 0 ą 0 such that for every k ě k 0 , every w P R k p,q pU qz TorpU q, and every z P X g w , the orbit closure L g pzq is ε-dense in X g w .

As a consequence, if K is a compact subset of B g , there is a real number pKq such that the length of every circle of every fiber X g w , for w P K, is bounded from below by pKq. Another consequence is: Lemma 5.3. For every δ 1 ą 0 there exists δ 2 ą 0 such that if w P B g is δ 2 -far from Critpπ g q, any circle in X g w escapes the δ 1 -neighborhood of Tangpπ g , π h q.

Let η denote half of the injectivity radius of the metric κ g . For every w P B g , the (riemannian) exponential map is a diffeomorphism from the disk of radius η in T w B g to some open subset of B g . By definition, the diameter of an interval J Ă T w B g is its length with respect to κ g,w ; its radius is half its diameter. Now, let I Ă B g be a smooth real analytic arc, and let w be a point of I. The tangent direction to I at w determines an orthogonal decomposition of T w B g into a direct sum T w I ' R pT w Iq K . Let r be a positive number ď η. We say that I is of size (at least) r at w if its preimage in T w B g by the exponential map contains the graph ts `ψpsq ; s P Ju of a function ψ : J Ă T w I Ñ pT w Iq K such that: (i) ψ is defined on an interval J of radius r around 0 in T w I; (ii) ψp0q " 0, its first derivative ψ 1 is bounded by 1, and its second derivative ψ 2 is bounded by 1{r.

Note that since ψ 1 is bounded by 1, ψ takes its values in an interval of length ď r in pT w Iq K . This definition is scale invariant. Similar notions can be defined on B h , or on X, or along the fibers X g w . When talking about the size of an arc, it should be clear from the context whether we are working in X, B g , or B h .

Since circles on a torus are geodesics for the flat metrics, we get:

Lemma 5.4. For every δ ą 0 there exists r 1 " r 1 pδq ą 0 such that if w P B g is such that distpw, Critpπ g qq ą δ, then any circle of X g w is of size at least r 1 at any of its points.

The ramification points of the restriction of π g to the leaves of the foliation induced by π h are located in Tang tt pπ g , π h q Y Tang ft pπ g , π h q Y Tang ff pπ g , π h q. This implies: Corollary 5.5. For every δ ą 0 there exists r 2 " r 2 pδq ą 0 such that if ξ P X is δ-far from Tang tt pπ g , π h q Y Tang ft pπ g , π h q Y Tang ff pπ g , π h q, then the projection under π g of any circle in X h π h pξq has size at least r 2 at π g pξq.

Of course a similar result holds by swapping g and h.

5.4.

Local structure of µ. Recall that we work under the hypotheses of Proposition 4.15.

By assumption, µ g pR g q is positive and µpTangpπ g , π h qq " 0, because the tangency locus Tangpπ g , π h q is a proper algebraic subset of X. And by Remark 4.13, µ g pR g q " µ h pR h q " 1.

Pick an analytic arc γ P R g such that µ g pγq ą 0; shrinking γ if necessary, we choose an open subset U as in Section 3.5, and parameters pα, βq and pp, qq such that γ is a smooth, real analytic subset of R α,β p,q pg; U q diffeomorphic to an interval. Then (5.2) 0 ă µpπ ´1 g pγqq " µpπ ´1 g pγq X π ´1 h pR h qq ď 1; consequently, there is an open subset U 1 of B h and a smooth analytic arc γ 1 in some R a 1 ,b 1 p 1 ,q 1 ph; U 1 q such that µpπ ´1 g pγq X π ´1 h pγ 1 qq ą 0. On the complement of Tangpπ g , π h q, the intersection π ´1 g pγq X π ´1 h pγ 1 q is transverse; thus reducing γ, U , γ 1 and U 1 again if necessary, we may assume that pπ g , π h q is a local diffeomorphism from X g U X X h U 1 to U ˆU 1 ; in particular the intersection π ´1 g pγq X π ´1 h pγ 1 q is transverse. Then, the set (5.3) π ´1 g pγq X π ´1 h pγ 1 q " S 1 \ ¨¨¨\ S l is a disjoint union of small "squares" -diffeomorphic to γ ˆγ1 -which, for w P γ, intersect the fiber X g w along pieces of circles (of the same slope). In what follows, we denote by S any one of the squares S j such that µpS j q ą 0; hence (5.4) S Ă π ´1 g pγq X π ´1 h pγ 1 q, µpSq ą 0, and pπ g , π h q : S Ñ γ ˆγ1 is a diffeomorphism. Thus, d R pµq ď 2, and Lemma 4.14 implies Lemma 5.6. d R pµq " 2.

We now initiate the study of the analytic continuation of γ. We say that an arc J Ă B g is evenly charged by µ g if µ g pJ 1 q ą 0 for every non-empty, open interval J 1 Ă J. Likewise, we say that a surface S in X is evenly charged by µ if µpS 1 q ą 0 for any non-empty relatively open subset S 1 Ă S.

For w 1 P γ 1 z TorpB hq and ξ 1 P X h w 1 , L h pξ 1 q is a finite union of circles of X h w 1 , and L 0 h pξ 1 q is invariant under some iterate of h. For µ h -almost every w 1 , the conditional measure µ h,w 1 is invariant under h w 1 ; as such, it is supported on a union of such orbit closures. The set S X X h w 1 is an interval, which is a piece of the circle L 0 h pξ 1 q, for ξ 1 P S X X h w 1 . The set 4(5.5) L h pSq "

ď ξ 1 PS L h pξ 1 q (resp. L 0 h pSq " ď ξ 1 PS L 0 h pξ 1 q ), π ´1 h pw 1 q B h γ 1 γ U 1 U S L 0 h pξq B g ξ w 1 w FIGURE 2.
This figure represents (one of) the square(s) S. The intersection of S with X h w 1 is contained in the orbit closure L 0 h pzq. The points ξ, w, w 1 are in green; the magenta points are critical values of π g and π h . The pre-image of γ in X h w 1 is locally contained in L 0 h pξq.

is a finite union of real analytic annuli -each of which a circle bundle above γ 1 -which is hinvariant (resp. is an h k -invariant annulus, for some k ą 0). For ξ 1 P S such that w 1 :" π h pξ 1 q R Tor h pB hq the restriction of the conditional measure µ h,w 1 to L 0 h pξ 1 q is the Haar measure. Projecting under the real analytic map π g , we get: Lemma 5.7. With the above notation, π g pL 0 h pSqq is a semi-analytic curve containing γ, which is evenly charged by µ g . In particular any (semi-)analytic continuation of γ contains π g pL 0 h pSqq.

Remark 5.8. Pick ξ 1 in S and set w 1 " π g pξ 1 q. If L 0 h pξ 1 q is disjoint from the ramification points of π g | X h w 1 : X h w 1 Ñ B g , then π g pL 0 h pSqq is an analytic loop in B g , and this loop provides a complete description of the analytic continuation of γ. On the other hand, if L 0 h pξ 1 q hits a ramification point of π g | X h w 1 , its image π g pL 0 h pξ 1 qq should be thought of as a segment which is strictly contained in the analytic continuation of γ, and whose endpoints are contained in the projections of the ramification points. As ξ 1 moves in S, the endpoints and the length of π g pL 0 h pξ 1 qq may vary within a curve contained in R g and after successively saturating as in (5.5), µ g might fill up a complicated, possibly dense, curve in B g . Lemma 5.9. The restriction of µ to S is absolutely continuous with respect to the 2-dimensional Lebesgue measure on S and its density with respect to any real analytic area form on S is given by a positive real analytic function.

The argument is the same as in Step 5 in the proof of Proposition 4.9. Now, from Section 5.3, we get the following a priori bound on the analytic continuation of γ. Lemma 5.10. For every δ ą 0 there exists r " rpδq ą 0 such that if γ P R g is an (analytic) arc with µ g pγq ą 0 and w is a point of γ such that distpw, Critpπ g qq ą δ, then γ admits an analytic continuation to an analytic arc of size r at w, which is evenly charged by µ g .

Proof. We may assume that w P γz TorpB g q. With notation as above, pick ξ P X g w X S. By Lemma 5.3, there exists ξ 1 P L 0 g pξq such that distpξ 1 , Tangpπ g , π h qq ą δ 2 " δ 2 pδq. By ginvariance of µ, ξ 1 plays the same role as ξ. Now by Corollary 5.5, π g pL 0 h pξ 1 qq has size r 2 pδ 2 q at π g pξ 1 q " w and applying Lemma 5.7 concludes the proof.

Remark 5.11. It may happen that the local equation for some R α,β p,q pB g q is of type Impw k q " 0, with w in a small disk D ε Ă B g (see the definition of NT g in § 3.4). At the origin of such a disk, the curve is singular, with several branches going through the origin. Lemma 5.10 says that if ε ! r and γ is a smooth analytic arc in one of these branches, then it can be continued to an evenly charged analytic arc accross the origin. So, if one of the branches is charged by µ, its symmetric with respect to the origin is charged too. 5.5. Analytic continuation of γ: the isotrivial case. From this point the proof splits in two separate arguments according to the isotrivial or non-isotrivial nature of the fibrations (more precisely, according to the emptyness, or not, of Tang tt pπ g , π h q). Lemma 5.12. If π g is isotrivial there is an analytic subset σ of B g such that:

(1) σ is of pure dimension 1 and extends γ;

(2) µ g pσq ą 1{2 and Supppµ g | σq is a semi-analytic set σ Ă σ; it is a finite union of immersed real analytic loops in B g and immersed real analytic arcs with endpoints in Critpπ g q.

Proof. As observed in § 5.2.2, π g becomes birationally equivalent to a trivial fibration after a finite base change B 1 Ñ B g . In particular its monodromy is finite and the curves R k p,q define global analytic subsets of B 1 . Coming back to π g : X Ñ B g , the local curves R k p,q pU q extend as (singular) global analytic subsets of B g . Since µ g pR g q " 1 we can find a finite number of smooth analytic arcs γ 1 , . . . , γ contained in R g such that µ g pγ 1 Y ¨¨¨Y γ q ą 1{2. Since every γ j is contained in some R k p,q pU q, it is contained in a global analytic subset σj of B g . We put σ " σ1 Y ¨¨¨Y σ . By Lemma 5.10, every irreducible component of σ X B g of positive mass is evenly charged by µ g . To conclude, we define σ to be the closure of the union of the components of σ X B g charged by µ. 5.6. Analytic continuation of γ: the general case. By Lemma 5.1, if Tang tt pπ g , π h q " H then π g and π h are isotrivial; in that case, Lemma 5.12 applies. Now, we assume Tang tt pπ g , π h q ‰ H, and our goal is to establish the following lemma. Lemma 5.13. If Tang tt pπ g , π h q ‰ H, then there exists a unique analytic curve σ g in B g such that (1) if µ is any ergodic Γ-invariant probability measure such that µ g pR g q ą 0, then µ g pσ g q " 1;

(2) if γ Ă σ g is any arc, then µpγq ą 0 for at least one such measure.

This result is both stronger and weaker than Lemma 5.12: indeed its conclusion holds for all invariant measures with µ g pR g q ą 0 (this fact will be important for Theorem C); on the other hand it gives no information on the structure of the analytic continuation σ g near Critpπ g q (this issue will be investigated in the next section).

The curve σ g is defined by this lemma. Note that, at this stage of the proof, it could contain for instance a sequence of small topological circles converging to a critical value of π g . We shall exclude such a possibility later. 5.6.1. An elementary lemma. Let M k : C Ñ C be the monomial M k pzq " z k . Lemma 5.14. Let r be a positive real number. For any 0 ă ε ! r there is a constant C r pεq ą 0 with the following property. If z 0 P C satisfies |z 0 | ď ε, and if γ is an analytic arc of size ě r at z 0 with 0 R γ, then M k pγq contains a point at which the curvature is ě C r pεqε ´k.

-l t 4 . * Proof. We prove it via an explicit computation; see Figure 5.6.1 for a graphical explanation. We may assume that r " 1, so that 0 ă ε ! 1, and that γ is of size 1 (we truncate γ if necessary). Let z 1 be the point closest to 0 on γ. Applying a rotation we may assume that z 1 " η P R for some η ď ε. Since z 1 is closest to the origin, γ is a graph over the y-axis: it is of the form x " ϕpyq, with ϕpyq " η `αy 2 `Opy 3 q. Since γ has size 1 at z 0 and ε ! 1, it has size ě 1{2 at z 1 , in particular |α| ď 2. In polar coordinates pρ, θq, the equation of γ is of the form (5.6) ρ " ψpθq " η ˆ1 `ˆ1 2 `αη ˙θ2 `Opθ 3 q ḟor θ small. Thus the polar equation of M k pγq near θ " 0 is

(5.7) ρ " ψpθ{kq k " η k ˆ1 `1 k ˆ1 2 `αη ˙θ2 `Opθ 3 q ˙,
and finally the value of the curvature at θ " 0 is

(5.8) ρ 2 `2pρ 1 q 2 ´ρρ 2 pρ 2 `pρ 1 q 2 q 3{2 ˇˇˇˇθ "0 " η ´k k ´1 ´2αη k ě η ´k ě ε ´k.
This completes the proof.

5.6.2. Extension of γ in B g : proof of Lemma 5.13. Let µ be an arbitrary Γ-invariant and ergodic measure, giving no mass to algebraic subsets, and assume that µ g pR g q ą 0. Then,by Remark 4.13, µ g gives full mass to R g .

Fix a small open disk U Ť B g in which we have fixed a section of π g and a continuous choice of basis for H 1 pX g w , Zq. In U , R g is the union of the analytic curves R k p,q pU q (see §3.5). To prove the lemma, we seek a uniform bound on maxpk, }pp, qq}q for the indices pk, pp, qqq such that µ g pR k p,q pU qq ą 0 for at least one Γ-invariant ergodic measure µ. Note that even for a single µ, µ g could charge infinitely many of the R k p,q pU q; due to the monodromy of the fibration, the analytic continuation of an arc γ of positive mass could come back infinitely many times in U , each time along a new R k p,q pU q. Suppose that there is a sequence of curves γ n " R kn pn,qn pU q with maxpk n , }pp n , q n q}q Ñ 8 such that µ n,g pγ n q ą 0 for some Γ-invariant measures µ n (that depend a priori on n). The γ n are evenly charged (by µ n ), and by Lemma 5.10 they have uniformly bounded geometry (the constant rpδq in Lemma 5.10 does not depend on µ n ). In particular the accumulation locus of pγ n q is uncountable.

Over U , there exists ε " εpk, p, qq, with εpk, p, qq Ñ 0 as maxpk, }pp, qq}q Ñ 8 such that for every w P U , any translate of L g,w pk, pp, qqq is ε-dense in X g w (see § 3.5 and Lemma 5.2). Hence, there is a sequence pε n q P pR ˚qN converging to 0 such that L g pξq is ε n -dense in X g w for any w P γ n z Tor g pU q and ξ P X g w . Since Tang tt pπ g , π h q is non-empty, it intersects every X g w along some non-empty finite subset, which by definition is not persistently contained in Singpπ h q. Furthermore Tang tt pπ g , π h q is generically transverse to π g . So we can pick w 0 in the accumulation set of pγ n q together with ξ t P Tang tt pπ g , π h q X X g w 0 such that π h pξ t q is 2δ-far from Critpπ h q for some δ ą 0, and ξ t can be locally holomorphically followed as a point ξ t pwq in Tang tt pπ g , π h q X X g w for w near w 0 . Pick a sequence w n P γ n z Tor g pU q converging to w 0 , and consider the corresponding sequence ξ t pw n q. For each n there is a finite union of annuli A n Ă X g γn such that π g maps A n onto γ n , π g : A n Ñ γ n is a fiber bundle whose fibers are unions of k n circles of type L 0 g pξq, and µ charges A n evenly. We can choose ξ 1 n P A n such that 0 ă distpξ 1 n , ξ t pw n q ă 2ε n . In X g wn the curve L 0 g pξ 1 n q is a circle, with uniformly bounded geometry. Near ξ t pw n q the restriction of π h to X g wn is locally conjugate to z Þ Ñ z k where k ě 2 is the order of tangency between π g and π h along Tang tt pπ h , π h q at ξ t pw n q. The local change of coordinates that transforms π h into z Þ Ñ z k depends on the fiber, hence on w n , but the first and second derivatives of these changes of coordinates are uniformly controlled, independently of n. Thus, Lemma 5.14 shows that the curvature of the projections π h pL 0 g pξ 1 n qq goes to `8 as n Ñ 8. But π h pL 0 g pξ 1 n qq is a piece of R h which is charged by µ h and is δ-far from Critpπ h q for large n; so by Lemma 5.10, its curvature must be uniformly bounded (with respect to n). This is a contradiction, and the proof of Lemma 5.13 is complete.

Remark 5.15. The proof shows that under the assumptions of Lemma 5.13, if γ is an arc of a curve R k p,q such that µ g pγq ą 0 for some invariant probability measure µ, then maxpk, }pp, qq}q is uniformly bounded. (5.9)

STang Γ " č ph 1 ,h 2 qPHalpΓq 2 pSingpπ h 1 q Y Singpπ h 2 q Y Tangpπ h 1 , π h 2 qq
By the Noether property, this infinite intersection can be written as a finite intersection. Thus, we can choose a finite set of Halphen twists pg i q 1ďiďs with g 1 " g and such that if x R STang Γ , there exist i ‰ j such that the fibers of π g i and π g j at x are smooth fibers and are transverse at x. For x R STang Γ , there are k ă and a (Zariski) neighborhood V Q x disjoint from Singpπ g k qXSingpπ g q such that π g k and π g are transverse in V . Let us show that in V , Supppµq is a real analytic set satisfying the conclusions of the proposition. There are two possibilities: (A1) either there exists k ‰ such that Tang tt pπ g k , π g q " H; then by Lemma 5.1 π g k and π g are isotrivial; (A2) or for every k ‰ , Tang tt pπ g k , π g q ‰ H. ' Let us first complete the proof in case (A1). Fix k ‰ given by Condition (A1) and apply Lemma 5.12: there exist analytic curves σg k in B g k and σg in B g such that µ g k pσ k q ą 1{2 and µ g pσ q ą 1{2. Then Σkl :" π ´1 g k pσ g k q X π ´1 g pσ g q is a real analytic subset of X such that µp Σkl q ą 0. Observe that Σkl is C-analytic in X (see § 5.1 for this notion): indeed the curves σg k and σg k , being of dimension 1, are defined by global equations in B g k and B g , hence so does Σkl " π ´1 g k pσ g k q X π ´1 g pσ g q on X. Let Σ0 be an irreducible component of Σkl such that µp Σ0 q ą 0. If f is an element of Γ, µpf p Σ0 qq " µp Σ0 q, and µpf p Σ0 q X Σ0 q " 0, unless f p Σ0 q X Σ0 contains a non-empty, relatively open subset. In the latter case f p Σ0 q " Σ0 . Thus, a finite index subgroup Γ 0 of Γ fixes Σ0 . Define (5.10)

Σ " ď f PΓ f p Σ0 q " ď f PΓ{Γ 0 f p Σ0 q
This is a C-analytic subset of X (hence of XzSTang Γ ) such that µp Σq " 1. We finally define Σ to be the union of the irreducible components of ΣzSTang Γ that are charged by µ. Using the Γ-invariance of XzSTang Γ and repeating the previous argument shows that Σ has finitely many irreducible components which are permuted by Γ (Assertion (2) of the proposition), and by construction Σ (hence Σ) is semi-analytic in X. By definition µpΣq " 1. Now choose any irreducible component Σ 1 of Σ and any x P Σ 1 . Around x, there are two transverse projections π g k and π g . The results of § § 5.3 and 5.4 imply that Σ 1 is locally a finite union of "squares" of the form π ´1 g pγq X π ´1 h pγ 1 q (where γ and γ 1 are smooth analytic curves), along which µ has a positive real analytic density. In particular µpΣ 1 q ą 0 and Σ 1 is evenly charged by µ. Thus, Assertions (1) and ( 4) of the proposition are established.

It remains to prove Assertion [START_REF] Blanc | On the inertia group of elliptic curves in the Cremona group of the plane[END_REF]. For this, note that if non-empty, the 1-dimensional part of SingpΣq is Γ-invariant. Suppose we can find an arc β in SingpΣq. If π g pβq were not a point, the closure of the g-orbit of β would contain a 2-dimensional annulus that would be contained in SingpΣq: contradiction. Thus π g pβq is a point, and so is π g 1 pβq for any g 1 P HalpΓq. This contradicts β Ă XzSTang 1 Γ . Thus, SingpΣq is a discrete subset of XzSTang 1 Γ . ' In case (A2), we fix k ‰ and apply Lemma 5.13 to π g k and π g : it yields analytic curves σ g k in B gk and σ g in B g such that µ g k pσ k q " 1 and µ g pσ q " 1. Set Σkl :" π ´1 g k pσ g k q X π ´1 g pσ g q; it is a 2-dimensional totally real C-analytic subset of XzpSingpπ g k q Y Singpπ g qq.

We then further restrict it to X k :" XzSTangpπ g k , π g qq and define Σ k to be the union of irreducible components of Σk X X k of positive µ-mass. Note that µpΣ k q " 1. From the analysis of § 5.4, we know that any irreducible component of Σ k is evenly charged by µ. So, for any other pair pk 1 , 1 q, the equality µpΣ k 1 1 X Σ k q " 1 implies that the analytic sets

Σ k and Σ k 1 1 coincide on X k X X k 1 1 . Thus the Σ k patch together to form a real analytic subset Σ of Ť k‰ X k " XzSTang Γ .
Since it is locally a finite union of 2-dimensional real analytic (and totally real) plaques, we infer that Σ is C-analytic in XzSTang Γ . Using the Γ-invariance of XzSTang Γ , we see that any component Σ 0 of positive mass is invariant under a finite index subgroup of Γ (see Equation (5.10) and the lines preceding it); hence, Σ has finitely many irreducible components. The remaining properties of Σ are obtained exactly as in case (A1), and the proof is complete.

Remark 5.16. The proof shows that if the curve σ g constructed in Lemma 5.13 is semi-analytic in B g , then the surface Σ is semi-analytic in X. This holds automatically in case (A1).

SEMI-ANALYTICITY OF Σ AND COMPLEMENTS

In this section we continue the investigation of case (c) of Theorem A by studying the semianalyticity of Σ. This leads to Theorem A' in § 6.3, and also prepares the ground for Theorem C. We keep the choice of Halphen twists g, h from § 5.2.1. By Remark 5.16 the semi-analyticity of Σ is already established in the isotrivial case, so: throughout this section we assume that Tang tt pπ g , π h q ‰ H. By Remark 5.16, we only need to show that the curve σ g from Lemma 5.13 admits a semianalytic continuation to B g . So, the work takes place near the singular fibers of π g . 6.1. Preparation and strategy. Recall that locally in B g , σ g is a union of smooth branches with uniformly bounded geometry. Recall also from § 5.1 that analytic curves have well-defined irreducible components. Lemma 6.1. Assume that Tang tt pπ g , π h q ‰ H. Any irreducible component of σ g is either an analytic loop in B g or an immersed line converging to Critpπ g q at its two endpoints.

Here by analytic loop we mean an analytic immersion of the circle, with possible selfintersections. And by an immersed line we mean an analytic immersion of the real line R Ñ B g .

Proof. Let σ be a component of σ g . If σ is compactly contained in B g then by Lemmas 5.10 and 5.13 it is an analytic curve in B g and we are in the first situation. Otherwise, there is a semiinfinite branch σ `of σ; since σ is analytic outside the finite set Critpπ g q and the accumulation set of σ `is connected, we deduce that the accumulation set of σ `is reduced to a singleton tc 0 u Ă Critpπ g q. Then the second branch must accumulate Critpπ g q as well (otherwise σ would be an analytic loop) and we are done.

From now on we study the structure of σ g locally near a fixed s P Critpπ g q. We already know that σ g is locally a union of smooth branches of the form R α,β p,q , with pα, β, p, qq P Q 2 {Z 2 ˆZ2 . The study of σ g will employ two types of arguments:

-first, we only use the dynamics of g and analyze the curves R α,β p,q near s, as started in Section 3. The rest of the group Γ is not taken into account. This allows us to make some operations (base change, blowing down p´1q-curves in fibers of π g , etc) which are not Γ-equivariant but preserve the curves R α,β p,q . This analysis, which is also crucial for Theorem C, is developed in §6.2.

-then in §6.3 we take into account the whole action of Γ on the singular fibers; doing so, we have to work on X, without simplifying the fibration π g . This is where condition (AC) enters into play.

6.2. Geometry of the curves R α,β p,q . As explained above, in this paragraph, the only information that we retain from the dynamics of Γ is the existence of the curve σ g , its geometric properties, and the fact that it is locally a union of smooth branches of the form R α,β p,q . After base change and some birational modification, as described in § § 3.3.3 and 3.3.4, which we simply refer to as the "reduction" of the singular fiber, we only have to consider a central fiber X g s of type I 0 (that is, a regular fiber) or I b with b ě 1.

6.2.1. Case 1: type I 0 . This corresponds to the stable reduction of fibers of type m I 0 , II, III, IV , I 0 , II ˚, III ˚, IV ˚and their blow-ups. Lemma 6.2. If the reduction of X g s is of type I 0 , then σ g admits a semi-analytic extension at s. In particular it admits only finitely many irreducible components in a neighborhood of s.

Proof. We may, and do assume that X g s is of type I 0 (but the results obtained so far for σ g hold only outside s).

Fix a pair of disks V Ť V 1 centered at s, with a local section of π g and a fixed basis for

H 1 pX g V 1 , Zq. Recall the real-analytic map T : V 1 Ñ R 2 introduced in §3.
2 and the definition (3.16) of R α,β p,q pV q in terms of T , which says that locally R α,β p,q pV q is the preimage of a straight line under T , in particular every branch of R α,β p,q pV qz tsu admits a semi-analytic extension at s. In V , σ g is a union of branches of R α,β p,q pV qz tsu. We need to show that σ g includes only finitely many such branches. For this, observe that by Lemma 5.13, σ g is analytic in V 1 zV . So if we can show that any irreducible component of σ g in V reaches BV , the finiteness follows. This relies on a topological argument that avoids explicit computations and will be used again below to prove Lemma 6.3. By Lemma 3.9 we may assume that either T is a diffeomorphism from V 1 to V 1 or that s is the unique critical point of T in V 1 . Suppose that some branch γ of R α,β p,q pV qz tsu is completely contained in V (see Figure 4 below): it is either an analytic loop in V z tsu or an immersed arc clustering at s at its two ends. We claim that γ contains a critical point of T , which is a contradiction. Indeed, parameterize γ by a smooth immersion ϕ : p0, 1q Ñ V , which extends continuously to r0, 1s, with ϕp0q " ϕp1q " c (with c " s if γ is not a loop). Then, T ˝ϕ is

T c γ V R α,β p,q (α, β) + R(p, q) FIGURE 4.
a smooth map from p0, 1q to the line pα, βq `Rpp, qq, it is continuous up to the boundary, and T ˝ϕp0q " T ˝ϕp1q; by Rolle's Theorem it admits a critical point in p0, 1q, as claimed.

6.2.2. Case 2: type I b , b ě 1. This corresponds to the stable reduction of fibers of type I b , m I b , I b (with b ě 1), and their blow-ups. As before we fix a pair of disks V Ť V 1 centered at s. The real analytic map T is only locally defined in V z tsu, but its critical points form a well defined subset of V 1 ztsu and, by Proposition 3.12, we can assume this set to be empty. We resume the computations from §3.6. From Equations (3.18) and (3.6), there is a coordinate w around s in which s " 0 and the equation of R α,β p,q is (6.1)

Im ˜tpwq ´`α `β b 2iπ log w p `q b 2iπ log w ¸" 0;
here log w is viewed as a multivalued function and tpwq is a well-defined holomorphic function near the origin (see § 3.3.2). As already observed, the curve R α,β p,q depends on pα, βq P Q 2 only through qα ´pβ P Q{Z; we denote by k the order of this torsion point qα ´pβ P Q{Z.

Lemma 6.3. If R α,β
p,q admits a compact component in V z tsu, then q " 0 and this component winds non-trivially around s.

Note that the notion of slope pp, qq depends a priori on our choice of local coordinates, but the property q " 0 does not: indeed the action of the monodromy around the loop is given by pp, qq Þ Ñ pp `bq, qq (geometrically, circles of slope p1, 0q correspond to the "vanishing cycle").

Proof. Let C be such a component. It can be parametrized by a degree 1 immersion ϕ : S 1 Ñ C (not necessarily injective, for C may have self-crossings). If C does not wind around the origin, there is a continuous determination of T ˝ϕ on S 1 . In this way, T ˝ϕ is a smooth map from S 1 to a line of slope pp, qq in R 2 , so it admits a critical point. This contradicts our choice of V 1 . So, the winding number C around the origin is not zero. Consider a small open disk U Ă V ztsu containing a point w 0 of C. If we follow the local determination of T along C by turning counterclockwise around the origin, T is composed with the monodromy pu, vq Þ Ñ pu ` C bv, vq. Thus, in U , C locally satisfies an equation of the form kT px, yq P Rpp, qq as well as kT px, yq P Rpp `n C bq, qq for every n P Z. Since C has only finitely many components in U , this implies q " 0. Lemma 6.4. Assume that the reduction of X g s is of type I b , b ě 1. Let C be a branch of σ g in V z tsu with q " 0 along some open subset of C. Then C admits a semi-analytic extension at s.

Proof. Since pp, qq is primitive, p " 1 and in some small disk U Ă V z tsu, C is of the form R α,β 1,0 . We can choose pα, βq of the form p0, βq, and the local equation of C becomes (6.2)

Im ˆtpwq ´β b 2iπ log w ˙" 0.
If β " 0 this defines an analytic subset of V . This set contains the origin if and only if the imaginary part ot tpwq vanishes at w " 0, in which case C coincides with a branch of tIm ptpwqq " 0u z t0u.

If β ‰ 0, we write w " e ´s as in §3.6, where s " x `iy ranges in some right half plane x ě x 0 . The Equation (6.2) rewrites as (6.3) Imptpe ´sqq ´βb 2π

x " 0, that is, tpx, yq ´βb 2π

x " 0, where tpx, yq is p2πq-periodic in y and admits a finite limit Imptp0qq as x Ñ `8; this defines an analytic curve C in the s-plane. In particular t is uniformly bounded along C, hence C is contained in a vertical strip tx 0 ď x ď x 1 u. The branch C is the projection under s Þ Ñ e ´s of a connected component C0 of C. It is contained in the annulus texpp´x 1 q ď |w| ď expp´x 0 qu and it is an analytic subset of this annulus because it is contained in σ g . According to Lemma 6.3, C must then be a loop that winds around the critical value s of π g . Remark 6.5. If Imptp0qq ‰ 0 and β ‰ 0 is small, Equation (6.3) defines a small loop around the origin. A priori, at this stage of the proof, σ g might have arbitrarily many small components of this type, converging to s. We shall rule out this phenomenon in Proposition 6.7.

Lemma 6.6. If the reduction of X g s is of type I b , b ě 1 and if C is a branch of σ g in V z tsu such that q ‰ 0 along some open subset of C, then one end of C converges to s and the other one escapes V . There are at most finitely many such branches in σ g .

Proof. The last statement follows from the first because every such branch reaches V 1 zV and σ g is analytic in V 1 zV .

Since q ‰ 0, we can choose pα, βq of the form pα, 0q for some α P Q, and the equation of C becomes (6.4) Im

˜tpwq ´α p `q b 2iπ log w ¸" 0
for some (primitive) slope pp, qq. The denominator of this expression does not vanish because |w| ă 1. Again, we shall write w " e ´s, s " x `iy, with x ą x 0 ě 0. By Lemma 6.3, C contains a branch that accumulates towards s. If the other branch escapes V , we are done. So, by Lemma 6.1, all we have to do is to show that the second branch does not converge towards s. We argue by contradiction, and parameterize C by an analytic immersion u P R Þ Ñ ϕpuq P V ztsu such that ϕpuq goes to s " 0 as u goes to `8 and ´8. Along that curve, the function u Þ Ñ tpϕpuqq converges towards tp0q as u goes to `8 and to ´8. Writing w " ρ expp2iπθq, Equation (3.6) yields 1 b τ pwq " θ ´i 2π logpρq; hence, the real part τ 1 of τ remains bounded while its imaginary part τ 2 goes to `8 along both ends of C. Now, let us consider the function T on the curve C. We start with a local definition of T in a small open subset U Ă V ztsu that intersects C; locally, T | C takes values in a line L Ă R 2 of slope pp, qq. Now, since R is simply connected, the function T | C can be analytically continued along C; since its values are locally contained in L, they are indefinitely contained in that line. Using Equation (3.3), we can write T " pt 1 ´τ1 τ 2 t 2 , 1 τ 2 t 2 q in U , and this local equality propagates along C by analytic continuation. Thus, T | C converges to pt 1 p0q, 0q at both ends. By Rolle's theorem, we deduce that the derivative of T | C vanishes at least once and, by Lemma 3.9, T has a critical point in V ztsu. This contradiction concludes the proof.

6.2.3.

A general finiteness result. The above results give a rather complete account of the geometry of the branches of σ g near a critical value of π g . However, for fibers of type I b , b ě 1, our results do not yet imply that σ g is semi-analytic at s: this is already apparent for the case of curves R α,β p,q with q ‰ 0 in the toy calculations of § 3.6. Still, we have the following finiteness result. It does not rely on the choice of a particular invariant measure, so it is stronger than the finiteness of the number of components of Σ in Proposition 4.15. This will play a key role in Theorem C. Proposition 6.7. If Tang tt pπ g , π h q ‰ H, then σ g has finitely many irreducible components.

Proof. Since σ g is analytic in B g , only finitely many components of σ g intersect any given compact K Ť B g . So, we can work locally near some fixed s P Critpπ g q. By Lemmas 6.2 and 6.6, the only case to deal with is that of fibers reducing to type I b and branches of type R k 1,0 (i.e. with q " 0). According to (the proof of) Lemma 6.4, we have to rule out the existence of an infinite sequence of small loops of the form R 0,βn 1,0 , with β n Ñ 0, winding around the origin s and converging to it (see Remark 6.5). For this we come back to the analysis of the local structure of invariant measures from § 5.4. If an arc γ Ă R 0,β 1,0 is locally contained in σ g in some open set U Ă V z tsu, there exists an ergodic invariant measure µ such that for w P γz Tor g pU q, its conditional µ g,w on X w g puts positive weight on some circle L 0 g pξq of slope p1, 0q through ξ (see § 5.4). Then, the g-invariance shows that µ g,w puts positive mass on each of the k components of L g pξq » L w pk, p1, 0qq, where k is the order of β in R{Z (see the definition of L w pk, pp, qqq in § 3.5.1 and the discussion around Equation (5.5) ). Now, since β n tends to 0, its order k n tends to infinity. Near s, this means that when σ g contains a component of type R 0,βn 1,0 , it contains also a component of R 0,jβn 1,0 for all j " 1, . . . , tβ ´1 n u. Fix δ ą 0 small and consider the level sets R 0,jβn 1,0 for δ ď jβ n ď 2δ. Then, from Lemma 6.4 (see Remark 6.5), this creates an accumulation of components of σ g away from Critpπ g q, which is a contradiction. 6.3. Condition (AC) and conclusion. If X g s is a singular fiber of π g with reduction of type I b , b ě 1, we define the active components of X g s to be the set of its irreducible components which are not contracted during the reduction process. More precisely, if X g s is not relatively minimal, there exists a unique birational morphism ε : X Ñ X 1 , with X 1 smooth, so that π g ˝ε´1 is relatively minimal: no fiber contains a p´1q-curve. By definition, the components of the exceptional divisor of ε are not active. After this contraction, X g s becomes a fiber εpX g s q of type m I b , for some m ě 1, or I b . In the first case the active components of X g s are the components which are not contracted by ε; equivalently, they are the components of the proper transform of εppX 1 q g s q. In the I b case, we retain only the proper transform of the b `1 components of multiplicity 2 in εppX 1 q g s q. The Active Components condition reads as follows:

(AC) there exists g P HalpΓq such that every fiber of π g reducing to type I b , b ě 1, contains an active component which is not in D Γ .

Theorem A'. If in Theorem A we further assume the non-degeneracy condition (AC), then in the totally real case pcq we can add the conclusion: (5) Σ is a semi-analytic subset of X.

Example 6.8. If D Γ is empty, then (AC) is satisfied. This is the case for general Wehler surfaces and general Enriques surfaces if one takes Γ " AutpXq (see [START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF] and the references therein). If X is minimal and π g does not contain singular fibers of type I b , then (AC) is satisfied.

Let us show that the examples obtained by Blanc's construction for three points (see § 2.2) satisfy condition (AC). One starts with a smooth cubic C Ă P 2 pCq and three general points p, q, r on C. Then, X is the blow-up of P 2 pCq at p, q, r, and at the 12 points appq, bppq, cppq, dppq, apqq, . . ., dpqq, aprq, . . ., dprq of C such that the tangent to C at one of these points intersects C in p, q, or r, respectively. The Jonquières involution s p (resp. s q , s r ) that preserves the pencil of lines through p (resp. q, r) and fixes C pointwise lifts to an automorphism of X. According to [START_REF] Blanc | On the inertia group of elliptic curves in the Cremona group of the plane[END_REF], the subgroup of AutpXq generated by s p , s q , s r is a free product Z{2Z ‹ Z{2Z ‹ Z{2Z. From the formulas given in [START_REF] Blanc | On the inertia group of elliptic curves in the Cremona group of the plane[END_REF]Lem. 17] for the action of s ‹ on NSpXq, it can be deduced that the composition g " s p ˝sq is a parabolic automorphism, preserving the pencil of plane quartic curves passing through p and q with multiplicity 2 and through the eight points appq, . . ., dpqq with multiplicity 1. The union of C and the line L p,q through p and q belongs to this pencil; this gives a reducible fiber of π g : X Ñ P 1 pCq which, after blowing down the strict transform of L p,q , is of type I 0 . It is easy to see that the unique effective divisor which is invariant by s p , s q and s r is the proper transform of C, i.e. D xsp,sq,sry " C. Since condition (AC) deals only with fibers of type I b , b ě 1, we conclude that it holds for xs p , s q , s r y. Example 6.9. General Coble surfaces with Γ " AutpXq do not satisfy condition (AC). More precisely, if g is a Halphen twist, then π g has 10 singular fibers with Identify V with D R , and for 0 ă R 1 ă R, consider the set π ´1 g pD R 1 q X Tang tt pπ g , π h q. As R 1 converges towards 0, this open subset of Tang tt pπ g , π h q converges towards the finite set Tang tt pπ g , π h q X X g s in the Hausdorff topology. Thus, if R 1 is small enough, π ´1 g pD R 1 q X Tang tt pπ g , π h q is a union of k subsets, each of diameter ď ε; its projection under π h is contained in a union of k disks ∆ i Ă B h , each of diameter less than Opεq.

Fix a sequence pw n q in γz TorpB g q converging towards s. For large n, pick a point ξ n P X g wn which is contained in a small square above γ on which µ restricts to a smooth measure, as in Section 5.4; the orbit L 0 g pξ n q is a circle in X g wn and by Lemma 6.10 a subsequence of L 0 g pξ n q converges to a logarithmic spiral in A. Changing ξ n into another point ξ 1 n P L 0 g pξ n q, we can therefore assume that (a) ξ n is in the support of µ, (b) w 1 n :" π h pξ 1 n q is δ-far from Critpπ h q Y Ť k i"1 ∆ i , and (c) w 1 n is not in TorpB h q; then, (d)

L 0 h pξ 1 n q is a circle in X h w 1
n , whose size at every point is bounded from below by a constant that does not depend on n (here we apply Lemma 5.4 to h).

The set π ´1 g pD R q X X h w 1

n is an open subset of X h w 1

n that contains ξ 1 n . If w 1 n is close enough to s and R is small, we may assume that the connected component

V 1 n of this open set that contains ξ 1 n contains a unique point s 1 of A, that V 1
n is a disk, and that

π g | V 1 : V 1 n Ñ V is
a covering which is ramified at s 1 only (indeed, property (b) above implies that Tang tt pπ g , π h q does not intersect V 1 ). Consider the connected component

I n Ă V 1 n of L 0 h pξ 1 n q that contains ξ 1 n .
The projection π g pI n q is locally contained in γ around w n . If I n did not contain s 1 , then π g | In would have no ramification point, so π g pI n q would be an arc with boundary points in BV ; being contained in this arc, γ would not converge to s, a contradiction. Thus I n contains s 1 and π g pI n q is a semianalytic subset of the disk V that contains s. It is smooth and analytic in V z tsu, and γ is a component of π g pI n qz tsu; therefore γ is semi-analytic, as desired.

FINITELY MANY INVARIANT MEASURES: PROOF OF THEOREM C

Let as usual X be a compact Kähler surface and Γ be a non-elementary subgroup of AutpXq containing a parabolic element. We want to show the following alternative:

-either pX, Γq is a Kummer example, -or there are only finitely many Γ-invariant ergodic measures with a Zariski dense support.

It is shown in § A.3 that Kummer groups can indeed admit infinitely many ergodic totally real measures (i.e. with d C pµq " 2 and d R pµq " 2).

Lemma 7.1. There is at most one Γ-invariant, ergodic probability measure which is absolutely continuous with respect to the Lebesgue measure.

Proof. Let µ and µ 1 be such measures. Fix a real analytic volume form ω on X, for instance ω " κ ^κ for some Fubini-Study form. From Theorem A, dµpxq " ξpxqω (resp. dµ 1 pxq " ξ 1 pxqω) for some function ξ (resp. ξ 1 ) which is positive and real analytic on the complement of some proper real analytic subset Bpµq (resp. Bpµ 1 q). On XzpBpµq Y Bpµ 1 qq, the function ξ 1 {ξ is continuous and Γ-invariant; since µ is ergodic, ξ 1 {ξ is constant, and µ 1 " µ.

According to Theorem A and Lemma 7.1, we are interested only in measures of type (c): those with a totally real support. To prove the finiteness we revisit the proof of Proposition 4.15 and use an alternative similar to that of §5.7; indeed, exactly one of the following two situations holds (compared with the alternative of §5.7, the quantifiers are switched): (A1') Tang tt pπ g , π h q " H for every pair g, h in HalpΓq such that π g ‰ π h (recall the convention of Remark 4.2); (A2') there exists g, h in HalpΓq such that Tang tt pπ g , π h q ‰ H.

Theorem C is then an immediate consequence of the following two lemmas. Lemma 7.2. If Alternative pA1 1 q holds, then pX, Γq is a Kummer group. Lemma 7.3. If Alternative pA2 1 q holds, then there are only finitely many ergodic, Γ-invariant probability measures with a Zariski dense support.

Proof of Lemma 7.2. By Proposition 3.12 in [START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF], we can choose g, h P HalpΓq which are conjugate in Γ and such that every periodic curve for h ˝g is contained in D Γ . With such a choice, the common components of π g and π h coincide with the components of D Γ , and the foliations F g and F h induced by π g and π h are everywhere transverse, except along Tang ff pπ g , π h q " D Γ .

Remark 7.4. If X Ñ X 0 is the contraction of D Γ (see Proposition 4.5), the fibrations π g and π h define two genus 1 fibrations on X 0 , and the associated foliations F 0 g and F 0 h are transverse everywhere, except on the finite set SingpX 0 q. Indeed, a smooth point of the surface can not be an isolated point of tangency between two foliations.

If β : r0, 1s Ñ B g is a path joining two points w 0 and w of B g , the holonomy of F h determines an isomorphism hol h pβq : X g w 0 Ñ X g w ; thus, π g is an isotrivial fibration (cf. Lemma 5.1). This construction defines a representation hol h : π 1 pB g ; w 0 q Ñ AutpX g w 0 q, the image of which fixes the finite subsets X h w 1 X X g w 0 for every w 1 P B h . Thus, hol h pπ 1 pB g ; w 0 qq is a finite subgroup H of automorphisms of the genus 1 curve E :" X g w 0 . A similar argument applies to π h in place of π g ; we shall denote by E 1 the genus 1 curve X h w 1 0 , for some w 1 0 in B h, and by H 1 the corresponding holonomy group. (Note that we have E 1 » E because the two fibrations are conjugate by some automorphism of X.) We also fix a point ξ 0 P X whose projections are w 0 " π g pξ 0 q and w 1 0 " π h pξ 0 q. This construction yields a holomorphic map Ψ from XzD Γ , or equivalently X 0 zSingpX 0 q, to E 1 {H 1 ˆE{H, which is defined as follows. To a point ξ in XzD Γ , we associate the intersection of the leaf of F g through ξ (i.e. X g πgpξq ) with the fiber X h w 1

0

; this gives a unique point modulo the action of H 1 , hence a point ψ 1 pξq P E 1 {H 1 . Doing the same with respect to F h and π g , we get a point ψpξq P E{H, and then we set Ψpξq " pψ 1 pξq, ψpξqq. Let ξ be a singularity of X 0 and let U be a small neighborhood of ξ. Then ψ 1 pX g πgpU q q is contained in a small disk V 1 Ă E 1 {H 1 ; similarly, ψpX h π h pU q q is contained in a small disk V Ă E{H. Thus, Ψ maps U ztξu in a bidisk V 1 ˆV ; as a consequence, Ψ extends to U , for the singularities of X 0 are normal (see [START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF]Prop. 3.9]). Altogether, this defines a finite ramified cover Ψ :

X 0 Ñ E 1 {H 1 ˆE{H.
We also define a regular map Φ from E 1 ˆE to X 0 . For this, without loss of generality we declare that the neutral element of X h w 1 0 is ξ 0 and denote by 0 the neutral element of E 1 ; hence, the pair pE 1 ˆt0u, p0, 0qq is identified to pX h w 1 0 , ξ 0 q by an isomorphism ϕ h : pE 1 , 0q Ñ pX h w 1 0 , ξ 0 q.

Similarly, we identify t0u ˆE to X g w 0 via an isomorphism ϕ g that maps 0 to ξ 0 . We shall denote by F 1 Ă E 1 (resp. F Ă E) the finite subset ϕ ´1 h pπ ´1 g pCritpπ g qq (resp. ϕ ´1 g pπ ´1 h pCritpπ h qqq); F 1 corresponds to the intersection of ϕ h pE 1 q with singular and multiple fibers of π g . If pu, vq is a point of E 1 ˆE close to p0, 0q, then the fibers X g πgpϕ h puqq and X h π h pϕgpvqq have a unique intersection point near ξ 0 . This defines a germ of diffeomorphism Φ : E 1 ˆE Ñ X 0 mapping p0, 0q to ξ 0 and preserving the fibrations; it is defined in a a small bidisk D 1 ˆD Ă E 1 Ê.

Observe that the composition Ψ ˝Φ coïncides with the natural projection from E 1 ˆE to E 1 {H 1 ˆE{H. Reducing the bidisk if necessary, this map Φ extends uniquely to D 1 ˆE and provides a local trivialization of the fibration π g above π g pD 1 q. Similarly, it extends to E 1 ˆD. So Φ is defined in a "cross" of the form D 1 ˆE Y E 1 ˆD. By analytic continuation, it extends uniquely to pE 1 zF 1 q ˆE and to E 1 ˆpEzF q, that is, to pE 1 ˆEqzpF 1 ˆF q. Moreover, Ψ ˝Φ : E 1 ˆE Ñ E 1 {H 1 ˆE{H is the quotient map with respect to the action of H 1 ˆH. From this, we deduce that the default of injectivity of Φ is given by a subgroup G of H 1 ˆH: Φppq " Φpp 1 q if and only if p 1 ´p P G. Since Ψ : X 0 Ñ E 1 {H 1 ˆE{H is a finite map, it follows that for each pu, vq P F 1 ˆF there is a point ξ P X 0 , an open neighborhood U of ξ, and an open neighborhood W of pu, vq such that Φ maps W ztpu, vqu into U . Embedding U into some affine space, we see by the Hartogs extension theorem that Φ extends to W . Thus, Φ extends to a holomorphic map E 1 ˆE Ñ X 0 which fits in a sequence (7.1)

E 1 ˆE Φ Ý Ñ X 0 Ψ Ý Ñ E 1 {H 1 ˆE{H,
such that the composition Ψ ˝Φ is the natural projection onto the quotient, and the fibers of Φ are orbits of G. Thus, X 0 is a generalized (singular) Kummer surface (see [START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF]): it is a quotient of the abelian surface E 1 ˆE by a finite subgroup G of H 1 ˆH; the singularities of X 0 correspond to the fixed points of elements of Gztidu. Restricting Φ to the complement of these fixed points, we get a regular finite cover onto the regular part of X 0 , with G as a group of deck transformations. Denote Λ and Λ 1 lattices in C such that E " C{Λ and E 1 " C{Λ 1 ; the universal cover of E 1 ˆE is C 2 , with projection C 2 Ñ C 2 {pΛ 1 ˆΛq. If we think of X 0 as an orbifold with quotient singularities, its universal cover is C 2 .

From this point, the argument is identical to [START_REF] Cantat | Finite orbits for large groups of automorphisms of projective surfaces[END_REF]Thm. 5.15]. Let f be an element of AutpX 0 q and lift it as a holomorphic diffeomorphism F of C 2 . Its differential DF px,yq at a point px, yq P C 2 is an element of GL 2 pCq. Let LpGq Ă GL 2 pCq be the linear part of G; it is a finite group, and the class rDF px,yq s of DF px,yq in GL 2 pCq{LpGq determines a holomorphic map that is invariant under translations by the lattice Λ 1 ˆΛ; since GL 2 pCq{LpGq is an affine variety, this map is constant. So, all lifts of all elements of AutpX 0 q to C 2 are affine maps. Now, consider the full group Γ Ă AutpXq. It preserves D Γ and induces a subgroup Γ 0 of AutpX 0 q. By the previous paragraph, all elements of Γ 0 come from affine transformations of E 1 ˆE. This proves that pX 0 , Γ 0 q and pX, Γq are Kummer groups.

Proof of Lemma 7.3. By Lemma 5.13, there is an analytic curve σ g Ă B g (resp. σ h Ă B h) such that µ g pσ g q " 1 (resp. µ h pσ h q " 1) for any ergodic invariant measure µ. By Proposition 6.7, there exists a compact subset K g Ť B g such that any component of σ g intersects K g (and similarly for h). In particular any invariant probability measure gives positive mass to π ´1 g pK g q X π ´1 g pK h q. In π ´1 g pK g q X π ´1 g pK h q, the regular set of π ´1 g pσ g q X π ´1 g pσ h q, which is semi-analytic, has finitely many connected components (see §5.1 and [2, Cor. 2.7]). By the analyticity of the density in Theorem A.(c), if Σ 0 is such a connected component, there is at most one ergodic invariant probability measure giving positive mass to Σ 0 , and the proof is complete.

INVARIANT ANALYTIC SURFACES WHICH ARE NOT REAL PARTS

In this section we construct examples of pairs pX, Γq such that Property (c) in Theorem A holds and for which:

(1) the support Σ of µ is an analytic real surface;

(2) there is no real structure on X for which Σ is contained in the real part XpRq. Λ " Z ' Zp 1 2 `itq " Z ' Zτ.

Since 1 2 ´it belongs to Λ, the complex conjugation z Þ Ñ z induces an anti-holomorphic involution σ E pzq " z on the elliptic curve E " C{Λ: this gives a real structure on E. The fixed point set of σ E gives the real part of E; one checks easily that it coincides with the projection of the real axis: (8.2)

EpRq " R{Z " R{pΛ X Rq.

8.2.

Abelian and Kummer surfaces. Now, consider the abelian surface A :" E ˆE " C 2 {Λ 2 , and the real structure σ A px, yq " px, yq mod pΛ 2 q. Its fixed point set is ApRq " EpRq 2 " R 2 {Z 2 . The group GL 2 pZq acts linearly on C 2 by preserving Λ 2 , so it also acts on A by "linear automorphisms". Set (8.3) e 1 " p1, 0q, e 2 " pτ, 0q, e 3 " p0, 1q, e 4 " p0, τ q.

The vectors e 1 and e 3 form a basis of the complex plane C 2 , and the four vectors e 1 , . . ., e 4 form a basis of the real vector space C 2 » R 4 . The real planes Vect R pe 1 , e 3 q and Vect R pe 2 , e 4 q are invariant under the action of GL 2 pZq, and they determine two invariant real tori in A, the first one being equal to ApRq. Define (8.4)

Σ A " Vect R pe 2 , e 4 q{Λ 2 " Vect R pe 2 , e 4 q{pZe 2 ' Ze 4 q » R 2 {Z 2 .
Now, consider the Kummer surface X 0 " A{η, where η is the holomorphic involution of A defined by ηpx, yq " p´x, ´yq mod pΛ 2 q. This surface has sixteen singularities, each of which is resolved by a simple blow-up. Let Σ 0 be the projection of Σ A in X 0 . We also let Σ X Ă X be the proper transform of Σ 0 in the smooth K3 surface X obtained by resolving the singularities of X 0 . 8.3. Statements. Theorem 8.1. With notation as above, the following properties are equivalent:

(a) there is a real structure s A on A whose real part contains Σ A ; (b) there is a real structure s X on X whose real part contains Σ X ; (c) the positive real number t is equal to ? 3{2, 1{2, or ? 3{6.

In addition when t " 1{2 (resp. ? 3{2 or ? 3{6) the curve E is isomorphic to the quotient of C by the lattice of Gaussian integers (resp. Eisenstein integers).

The equivalences (a)ô(c) and (b)ô(c) are proven respectively in § 8.4 and 8.5 below.

Corollary 8.2.

There exist examples of abelian and Kummer surfaces X, and non-elementary subgroups Γ Ă AutpXq such that (1) Γ preserves an analytic, totally real surface Σ Ă X, (2) Γ has a dense orbit and an invariant, ergodic, smooth, probability measure with support equal to Σ, and (3) there is no real structure on X whose real part contains Σ.

Indeed, one can just take Γ " GL 2 pZq in the previous examples, and for the invariant probability measure one takes the measure coming from the Lebesgue measure on Σ A . 8.4. Proof for the abelian surface. Let s be a real structure on A, i.e. an anti-holomorphic involution. Then s ˝σA is holomorphic, so s " B ˝σA for some automorphism B of A. Now, assume that the fixed point set of s is equal to Σ A . Since the origin p0, 0q of A is fixed by σ A and s, it is fixed by B too. This means that B is induced by a linear automorphism of the complex vector space C 2 , i.e. by an element of GL 2 pCq.

Near the origin of A, we can write spx, yq " B ˝σA px, yq " Bpx, yq, and our assumption implies that the vector e 2 " pτ, 0q satisfies Bpτ , 0q " pτ, 0q, i.e. Be 2 " pτ {τ qe 2 because B is C-linear. The same property is satisfied by e 4 . Since pe 2 , e 4 q is a basis of the complex plane C 2 , B is a homothety: B " pτ {τ qId. As a consequence, the linear map M : C Ñ C defined by M pzq " pτ {τ qz preserves the lattice Λ " Z ' Zτ , with τ " 1{2 `it. Thus, we can find a quadruple of integers pa, b, c, dq such that (8.5) τ τ " a `bτ and τ τ τ " c `dτ.

This implies a " c " ´1, by looking at the imaginary parts after both equations have been multiplied by τ . Then (8.6)

1 " b|τ | 2 and 3 4 ´t2 " d|τ | 2 .
The first equation plus the relation |τ | 2 " 1{4 `t2 ą 1{4 imply that 1 ď b ď 3, and more precisely pb, tq P tp1, ? 3{2q, p2, 1{2q, p3, ? 3{6qu. Together with the second equation we end up with exactly three possibilities:

(1) pb, d, tq " p1, 0, ? 3{2q, the lattice Z'Zτ is the lattice of Eisenstein integers Zrp1`i ? 3q{2s, and spx, yq " pe 2iπ{3 x, e 2iπ{3 xq " e 2iπ{3 σ A px, yq;

(2) pb, d, tq " p2, 1, 1{2q, τ " p1 `iq{2, the lattice is the image of the lattice of Gaussian integers Zris by the similitude z Þ Ñ 1´i 2 z, and spx, yq " pix, iyq " iσ A px, yq; (3) pb, d, tq " p3, 2, ? 3{6q and τ " 1{2 `i? 3{6; modulo the action of PSL 2 pZq on the upper half plane, τ is equivalent to 2 ´1{τ " p1 `i? 3q{2 so we end up again with the Eisenstein integers, and spx, yq " pe 2iπ{6 x, e 2iπ{6 yq " e 2iπ{6 σ A px, yq This completes the proof of the implication (a)ñ(c), and for the converse implication the explicit formulas for spx, yq provides the desired real structure.

Note that in each of these cases, s is conjugate to σ A by β id P GL 2 pCq, with respectively β " e 2iπ{6 , e 2iπ{8 , e 2iπ{12 . However in the second and third cases, this conjugacy is only satisfied near the origin, because β id does not preserve the lattice (i.e. it does not induce an automorphism of A). 8.5. Proof for the (smooth) Kummer surface. Suppose there is a real structure s X on X whose fixed point set contains Σ X . Let σ X be the real structure induced by σ A on the Kummer surface X. Then, there is an automorphism B X of X such that s X " B X ˝σX .

Consider the origin p0, 0q of A, and its blow-up ε : Â Ñ A. In local coordinates it expresses as εpu, vq " pu, uvq " px, yq P A, with exceptional divisor D " tu " 0u. The involution η lifts to ηpu, vq " p´u, vq; it is the identity on D, it acts transversally as u Þ Ñ ´u, and the quotient map Â Ñ X " Â{η is locally given by q : pu, vq Þ Ñ pu 2 , vq. Lifting σ A , we obtain σA pu, vq " pu, vq. In A, Σ A is locally parametrized by psτ, s 1 τ q with s and s 1 small real numbers; its strict transform is the real analytic surface ΣA given by pu, vq " psτ, s 1 {sq. So, the intersection of ΣA with D is determined by the equation v P R. In particular, ΣA X D is fixed by σA . The image of D in X is a curve C » P 1 pCq and the image of the subset tu " 0, v P Ru is a great circle S Ă C. This circle is fixed by σ X , as we just saw, and by s X , by definition of Σ X . Thus, B X fixes S, hence C itself since S is Zariski dense in C (for the complex algebraic structure on X).

Since B X fixes C, we can contract C onto a singularity of X 0 : B X and σ X descend to regular (holomorphic and anti-holomorphic) maps on a neighborhood of the singularity. Since this singularity is the quotient singularity pC 2 , 0q{η, we can lift B X , σ X , and s X to germs of diffeomorphisms B A , σ A , and s A near the origin in C 2 . Using the natural, local coordinates given by the projection C 2 Ñ A Ñ X 0 , the lifts can be written (8.7) σ A px, yq " px, yq and B A px, yq " p

ÿ k, ě0 a k, x k y , ÿ k, ě0 b k, x k y q
for some locally convergent power series ř k, a k, x k y and ř k, b k, x k y . In these coordinates, Σ X corresponds to the real plane puτ, vτ q for pu, vq P R 2 , and the equation for the fixed points of s gives B A puτ , vτ q " puτ, vτ q for pu, vq P R 2 . This implies that B A is linear in these coordinates, equal to the homothety of factor τ {τ .

Up to this point, we have worked locally near the singularity of X 0 corresponding to the origin of A, we now globalize the argument. As a consequence of its local form, B A preserves the horizontal line ty " 0u, so that B X preserves the quotient curve (8.8) pC{pΛq ˆt0uq{η » P 1 pCq Ă X.

Furthermore, in the coordinate x given by the projection (8.9) C ˆt0u Ñ E ˆt0u " C{Λ ˆt0u Ñ P 1 pCq (where the last arrow is a branched cover of degree 2), B X is covered by the linear map x Þ Ñ pτ {τ qx. Thus the analysis of the previous subsection applies, and shows that τ " 1{2 `i? 3{2, p1 `iq{2, or 1{2 `i? 3{6, and the proof of (b)ñ(c) complete. For the converse implication, it is enough to observe that for each of these three cases, the explicit anti-holomorphic involution on A given in § 8.4 commutes with η, so it descends to the Kummer surface X.

INVARIANT SURFACES WITH BOUNDARY

In this section, we show that in case (c) of Theorem A, the surface Σ may have a non-trivial boundary in X. We provide two examples, one for a Kummer surface, and then a deformation keeping the main features of the first example but on a surface which is not anymore a Kummer surface. We also give examples of invariant curves that do not support any invariant measure. 9.1. On a Kummer surface. Consider a Kummer example, with the same construction as in Sections 8.1 and 8.2. Embed the curve E " C{Λ in P 2 , in a Weierstrass form. Its equation is (9.1) y 2 " 4x 3 ´g2 x ´g3

with coefficients g i P R depending on the parameter t; the real structure σ E pzq " z is the restriction to E of the real structure rx : y : zs Þ Ñ rx : y : zs on P 2 . Since EpRq " Fixpσ E q is connected, g 2 is negative. By convention, we fix the origin of the elliptic curve for its group law at the (inflexion) point at infinity. If u and v are two points of E, the line containing u and v intersects E in a third point w. The sum u`v`w is zero for the group law. If u " px, yq is a point of E, then ´u " px, ´yq and the fixed points of this involution u Þ Ñ ´u on EpRq are the two points px 0 , 0q and p8, 8q where x 0 is the unique real solution of the equation 4x 3 " g 2 x `g3 . Now, consider the map Φ : E ˆE Ñ P 1 ˆP1 ˆP1 which is defined as follows: if pu, vq belongs to E ˆE, with u " px 1 , y 1 q and v " px 2 , y 2 q, and if w " ´pu `vq " px 3 , y 3 q, then Φpu, vq " px 1 , x 2 , x 3 q. One easily checks that Φ ˝η " Φ, with ηpu, vq " p´u, ´vq, and Φ embeds the Kummer surface X 0 " pE ˆEq{η into P 1 ˆP1 ˆP1 as a singular p2, 2, 2q-surface (see [7, §8.2]). The singularities of X 0 correspond to the fixed points of η, i.e. to the group Ar2s of torsion points of order 2 in A " E ˆE. This gives 16 points, of which only 4 are real: (9.2) px 0 , x 0 , 8q, px 0 , 8, x 0 q, p8, x 0 , x 0 q, p8, 8, 8q.

The real part of X 0 corresponds to the fixed point set of σ 0 , i.e. of σ A viewed on the quotient space X 0 . Recall that σ A pz 1 , z 2 q " pz 1 , z 2 q if we think of A as C 2 {Λ 2 . The fixed points of σ 0 are of two types: those coming from the fixed points of σ A , hence from the real part ApRq, and those coming from (9.3) ∆ :" tpz 1 , z 2 q P A ; σ A pz 1 , z 2 q " ηpz 1 , z 2 qu.

In the quotient X 0 , ApRq projects onto a sphere with four singularities; the projection of ∆ is another sphere with the same four singularities. These two spheres are glued along those four points; locally X 0 is the quotient of C 2 by tid, ´idu, so up to an analytic change of coordinates, X 0 pRq is a quadratic cone isomorphic to x 1 x 2 " x 2 3 .

X 0 pRq

The natural action of GL 2 pZq on E ˆE descends to an action of PGL 2 pZq on X 0 , which preserves X 0 pRq; it also preserves individually each of the two connected components of X 0 pRqzSingpX 0 pRqq. The action of PGL 2 pZq on these two punctured spheres has dense orbits (and finite orbits too, corresponding to torsion points of A). If we resolve the singularities of X 0 pRq, the two punctured spheres become two surfaces homeomorphic to a sphere minus four disks; they are glued together along their boundaries to form a closed, orientable surface of genus 3, which is the real part of XpRq for the real structure σ X . Thus, the generic orbit of PGL 2 pZq in XpRq is dense in one of these two open subsets of XpRq. This gives a first example of an invariant surface with boundary Σ, given by the component ApRq{η, with an invariant measure µ given by the push forward of the Haar measure from ApRq to Σ. 9.2. Deformation. Let us come back to X 0 pRq. Switching the chart in P 1 ˆP1 ˆP1 so that the coordinates px 1 , x 2 , x 3 q are replaced by their inverses p1{x 1 , 1{x 2 , 1{x 3 q, the four singularities become (9.4) pα, α, 0q, pα, 0, αq, p0, α, αq, p0, 0, 0q with α " 1{x 0 . Note that the three vectors v 1 " pα, α, 0q, v 2 " pα, 0, αq, and v 3 " p0, α, αq are linearly independent (their determinant is ´2α 3 ). Thus, given any triple pε 1 , ε 2 , ε 3 q P t˘1u 3 , there is a real quadratic form Qpx 1 , x 2 , x 3 q such that ε i Qpv i q ą 0 for each 1 ď i ď 3. If P denotes the equation of X 0 (after changing the x i in 1{x i as above) and ε is a small real number, then P `εQ is an equation of a new surface X ε of degree p2, 2, 2q in P 1 ˆP1 ˆP1 . At the origin p0, 0, 0q, the linear term of the equation P " 0 is not changed by the addition of εQ, and the quadratic term is only slightly perturbed if ε is small enough, so X ε still admits a quadratic singularity, which is non-degenerate of signature p2, 1q. At the other three real singularities of X 0 , we can choose the sign of Qpv i q in such a way that X ε pRq is locally disconnected (as does a hyperboloid with two sheets). We claim that, shifting Q a little bit if necessary, all (real or complex) singularities of X 0 disappear in the perturbation X ε , except the origin. Indeed, by conjugating by a diagonal automorphism ph, h, hq P AutpP 1 q 3 , such that h P PGL 2 pCq fixes 0 and α, we can arrange that all singular points of X belong to C 3 . Let ps j q j"0,...,15 be the singular points of X 0 (with s 0 " 0) and choose Q in such a way that Qps j q ‰ 0 for j ě 1. Let N " Ť j Bps j , ηq, where η is so small that Q ‰ 0 on Ť jě1 Bps j , ηq. Take ε to be small and non-zero. Then,X ε is smooth in Ť jě1 Bps j , ηq, because its equation P `εQ " 0 reduces to P {Q `ε " 0 there, and such a hypersurface is smooth for ε ‰ 0 small. Finally, smoothness being an open property, X ε is also smooth in the complement of N .

S

Xε pRq Remark 9.1. Such a deformation appears naturally in the closely related example of character varieties for the once punctured torus (see [START_REF] Cantat | Painlevé and Schrödinger[END_REF][START_REF] Goldman | Topological components of spaces of representations[END_REF]).

The result, for a sufficiently small ε and a good choice of Q, is a real surface X ε of degree p2, 2, 2q in P 1 ˆP1 ˆP1 with a real part X ε pRq satisfying the following properties.

-The surface X ε pRq has a unique singularity, at the origin.

-After a minimal resolution of the singularity, we get a real K3 surface Xε . Indeed, the area form Ω Xε defined in Example 2.1 lifts to a trivialization of the canonical bundle of Xε ; and Xε pCq is simply connected (to see this one can use the fibrations πi : Xε pCq Ñ X ε pCq Ñ P 1 pCq). -The surface Xε pRq is homeomorphic to a sphere S 2 ; the exceptional divisor is a curve S ε Ă Xε pRq 0 that separates Xε pRq in two one-holed spheres with boundary S ε ; -The three involutions σ i : X ε Ñ X ε described in Example 2.1 lift to three automorphisms of Xε . They generate a non-elementary subgroup Γ ε of Autp Xε q that preserves the real structure (the compositions σ i ˝σj are parabolic automorphisms with respect to distinct fibrations). A subgroup of index 4 preserves simultaneously each component of Xε pRqzS and the canonical area form of the K3 surface.

This provides examples with non-trivial invariant open subsets of XpRq for a non-elementary subgroup of AutpX R q, in a case where X is not a Kummer surface (the dynamics of the group Γ ε is not covered by a linear dynamics on a torus). 9.3. Invariant curves. Let us continue with the example given by X ε . When ε " 0, X 0 is a (singular) Kummer surface, and its singularities are in 1 to 1 correspondance with the element of Ar2s. Let Γp2q Ă GL 2 pZq be the subgroup that fixes Ar2s pointwise. The image of Γp2q in AutpX 0 q preserves the 16 singularities of X 0 . This group lifts to a group of automorphisms Γ 0 Ă Autp X0 q. Let S 0 Ă X0 be the p´2q-curve obtained by the minimal resolution of one of the singularities of X 0 . The dynamics of Γ on S 0 coincides, up to conjugacy, with the linear projective action of Γp2q Ă GL 2 pZq on PpC 2 q. This is a non-elementary subgroup of AutpP 1 q » PGL 2 pCq; in particular, this action does not preserve any probability measure. Now, consider the small perturbation Xε and the group Γ ε , as constructed in Section 9.2. Then, Γ ε preserves the p´2q-curve S ε Ă Xε , and for ε " 0, we recover the group Γ 0 up to finite index. Since the non-elementary property of Γ 0 | S 0 Ă PGL 2 pCq is invariant under small perturbations or after taking finite index subgroups, we deduce that Γ ε induces a non-elementary subgroup of S ε . Thus, we obtain examples of K3 surfaces Xε with of a non-elementary subgroup Γ ε Ă Autp Xε q such that Γ preserves a smooth rational curve S ε Ă Xε but S ε does not support any Γ ε -invariant probability measure; here, the examples are deformations of a Kummer example p X0 , Γ 0 q. Remark 9.2. Similar examples can be constructed on some Coble surfaces Y : AutpY q preserves a rational curve, coming from a plane sextic with ten nodes; and then, taking a double cover of Y ramified along the invariant sextic, one gets K3 surfaces. Remark 9.3. Consider the above example p Xε , S ε , Γ ε q and a probability measure ν on Γ ε whose support is finite and generates Γ ε . Then, the curve S ε » P 1 pCq supports a unique ν-stationary measure µ ν , because Γ ε Ă PGL 2 pCq is non-elementary (see [START_REF] Furstenberg | Noncommuting random products[END_REF]). If, as above, everything is defined over R, the support of µ ν is contained in a circle; but if we apply the same construction with a well chosen, small complex deformation X ε , the support of µ ν is supported on a fractal quasi-circle.

APPENDIX A. ABELIAN SURFACES

In this appendix, we consider the case when all parabolic automorphisms g of Γ induce an automorphism g B of infinite order on the base of their invariant fibration π g . In that case, we know from [START_REF] Cantat | Symétries birationnelles des surfaces feuilletées[END_REF]Proposition 3.6] that X is a compact torus, and in fact an abelian surface since Γ is non-elementary. Thus, we assume that (i) X is an abelian surface, isomorphic to C 2 {Λ for some lattice Λ; Lemma A.5. If there is a parabolic element g P Γ for which pp g q ˚µqpRpgqq ă 1, then µ is the Haar measure on X.

Proof. The proof is the same as for Proposition 4.9. Pick another parabolic transformation h P Γ, such that g and h are linearly independent; such an h exist because Γ is non-elementary. The main tool is the disintegration of µ with respect to g ; for y in a subset Y g Ă B g {T g of positive measure, the conditional measure λ g,y is the Haar measure on the 3-dimensional torus ´1 g pyq. Hence dim R pµq ě 3 and pp g q ˚µqpRpgqq " 0, as in Step 2 of the proof of Proposition 4.9. As in Steps 3 and 4, we infer that p h q ˚µ does not charge Rphq and that p h q ˚µ is absolutely continuous with respect to the Lebesgue measure on R{Z. This, implies that µ itself is invariant by all translations along the fibers of h , because µ " ş Y λ h,y dpp h q ˚µqpyq and λ h,y is the Haar measure for almost every y. Permuting the roles of g and h, µ is in fact invariant under all translations. Hence, µ is the Haar measure on X. Now, we we are reduced to the case where p g q ˚µpRpgqq " 1 for every parabolic automorphism g in Γ. Since Rpgq is countable, d R pµq ď 3 and µ charges some fiber of g . Using another parabolic automorphism h, we see that µ gives positive mass to a translate a 0 `S0 of a 2-dimensional torus S 0 Ă X whose projections g pa 0 `S0 q " g pa 0 q and h pa 0 q are in the countable sets Rpgq Ă B g {T g and Rphq Ă B h {T h respectively. Thus, by ergodicity, we conclude that µ is supported on a finite union of translates of 2-dimensional tori a j `Sj Ă X, 0 ď i ď k ´1 for some k ě 1.

A subgroup Γ 0 of index ď k! in Γ preserves a 0 `S0 , and g k! and h k! act on a 0 `S0 » R 2 {Z 2 as two linear parabolic transformations with respect to transverse linear fibrations. So, it follows that µ |a 0 `S0 is proportional to the Haar measure of a 0 `S0 , and the proof of Proposition A.1 is complete.

A.3. No or infinitely many invariant real tori. Consider a compact complex torus X " C 2 {Λ of dimension 2. Let Γ be a subgroup of AutpXq. As in § A.1, write the elements f of AutpXq in the form f px, yq " A f px, yq `Sf , and denote by A Γ Ă GL 2 pCq the image of Γ by the homomorphism f Þ Ñ A f . The group Γ is non-elementary if and only if A Γ contains a free group, if and only if the Zariski closure of A Γ in the real algebraic group GL 2 pCq is semisimple. Now, assume that Γ is non-elementary and preserves at least one ergodic probability measure µ with dim R pµq " 2. Equivalently, after conjugation by a translation, there is a finite index subgroup Γ 0 Ă Γ that preserves a real, two-dimensional subtorus Σ " Π{Λ Π , where Π Ă C 2 is a real vector space of dimension 2 and Λ R :" Π X Λ is a lattice in Π (the restriction of µ to Π{Λ Π is proportional to the Haar measure). The goal of this last section is to explain that, in fact, Γ preserves infinitely many ergodic measures µ j with dim R pµ j q " 2. Two mechanisms can be used to establish this fact.

The first one relies on the fact that Γ 0 acts on the quotient Q " X{Σ » R 2 {Z 2 , fixing the origin. Moreover, the action of Γ 0 on Q " R 2 {Z 2 is induced by an injective homomorphism Γ 0 Ñ GL 2 pZq (to see this, note that C 2 " Π ' R iΠ and iΠ surjects onto Q). This implies that Γ 0 has arbitrarily large finite orbits in Q (coming from torsion points of Q). The preimages of these orbits in X provide surfaces Σ j Ă X; they are "parallel" to Σ and have an arbitrarily large number of connected components; they are Γ 0 -invariant; and each of them supports a unique invariant, ergodic, probability measure µ j with dim R pµ j q " 2.

For the second mechanism, we assume that Γ 0 fixes the origin and, changing Γ 0 in a finite index subgroup if necessary, we identify Γ 0 with a subgroup of SL 2 pCq. Identify pΠ, Λ Π q to pR 2 , Z 2 q and the restriction Γ 0 | Π to a subgroup of GL 2 pZq; since Γ 0 is non-elementary, Γ 0 is Zariski dense in SL 2 pCq and Γ 0 | Π is Zariski dense in SL 2 pRq (resp. in SL 2 pCq). In particular, the Q-algebra generated by Γ 0 | Π is the algebra of 2 ˆ2 matrices with rational coefficients. The decomposition C 2 " Π ' R iΠ is Γ 0 -invariant, and the multiplication by i defines a Γ 0equivariant map from Π to iΠ. Thus, Γ 0 preserves each of the real planes Π η " tpx, yq ὴipx, yq ; for px, yq P Πu, with η P R. Now, consider the (real) projection q of C 2 onto Π parallel to iΠ, and set Λ 1 " qpΛq. It is a Γ 0 -invariant subgroup of Π of rank at most 4, and it contains Λ Π » Z 2 . Then, one checks easily that (1) Λ 1 is commensurable to Λ Π ' αΛ Π , for some α P RzQ, or to Λ Π , in which case we set α " 0; (2) Λ is commensurable to Λ Π ' K α,β pΛ Π q where K α,β is the linear map from Π to Π ' iΠ defined by K α,β puq " αu `βiu; (3) for m in Z, the real plane Π mα{β is Γ-invariant and intersects Λ on a cocompact lattice Λ Π mα{β .

Then, the surfaces Σ m " Π mα{β {Λ Π mα{β form an infinite family of Γ-invariant tori in X.

Remark A.6. This second argument does not apply in the following case. Let E " C{Zris, Λ " Zris ˆZris Ă C 2 , and X " C 2 {Λ " E ˆE. The group Γ " SL 2 pZq ˙R2 {Z 2 is a subgroup of AutpXq that preserves the torus Π{Λ Π for Π " R 2 Ă C 2 , but has no fixed point (because Γ contains Π{Λ Π ), and every Γ-invariant surface is a finite union of translates of this torus .

On the other hand, this second argument applies when X and Γ come from a genuine Kummer example, that is, a Kummer example defined on a surface that is not a compact torus. Indeed in that case Γ contains a finite index subgroup with a fixed point.

1. 5 .

 5 Structure of the paper. In Section 2 we start by briefly describing a few basic examples which are useful to be kept in mind; more advanced examples are given in Sections 8 and 9.

Example 4 . 6 . 4 . 2 .

 4642 The Coble surfaces and the surfaces constructed by Blanc (see §2.2) provide examples of pairs pX, Γq such that X is rational and Γ preserves a smooth rational curve or a smooth curve of genus 1, respectively. Analytic subsets of positive mass, d R pµq, and d C pµq. Let µ be a Γ-invariant and ergodic probability measure. Denote by d R pµq the minimum of the dimensions k P t0, 1, 2, 3, 4u such that there exist an open set U Ă X, for the euclidean topology, and a real analytic submanifold W Ă U of real dimension k with µpW q ą 0.

  g,w pzq dµ g pwq for every Borel function ξ : X Ñ R. The measures µ g,w are unique, in the following sense: if µ 1

4. 4 .

 4 Dimension ď 1. Lemma 4.14. If d R pµq ď 1 then µ is either supported on a finite orbit, or on D Γ .

FIGURE 3 .

 3 FIGURE 3. A (green) curve of bounded geometry passing near the origin and its image (in blue) by z Þ Ñ z 2 on the left, and z Þ Ñ z 5 on the right.

5. 7 .

 7 Conclusion of the proof of Proposition 4.15. Proof of Proposition 4.15. Let g be as in the statement of the proposition. Recall the definition of STang Γ from (4.4):

8. 1 .

 1 A family of lattices. Let t be a positive real number, and set τ " 1 2 `it, where i " ?´1. Consider the lattice Λ Ă C defined by(8.1)

  ). An example of a different kind is given in Section 8.A noteworthy consequence of Propositions 4.15 and 4.9 is: Corollary 4.17. The dimension d R pµq cannot be equal to 3. 4.6. Proof of the main theorem, and consequences. Proof of Theorem A. If µ gives positive mass to a proper algebraic subset of X, then according to Lemma 4.8, either Assertion (a) or (b) of Theorem A is satisfied. Otherwise, we know from Lemma 4.14 and Corollary 4.17 that d R pµq is either equal to 2 or 4, and exactly one of Proposition 4.15 or Proposition 4.9 applies. If d R pµq " 2, Proposition 4.15 shows that the conclusions of case (c) of the theorem hold. The remaining case (d) is covered by Proposition 4.9. In both cases the exceptional set Z is equal to STang Γ .

Proof of Corollary B. When Γ does not preserve any proper algebraic subset, then we must be in one of the cases (c

) (if d R pµq " 2) or (d) (if d R pµq " 4) of Theorem A. Furthermore STang Γ is empty. Thus if d R

pµq " 4 we infer that µ has an analytic positive density on X. If d R pµq " 2, note that the analytic surface Σ is smooth everywhere, since otherwise its singular locus would be a finite invariant subset (see Assertion (4) of Proposition 4.15).

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

We must take the closure in Equation

5.5 because L 0 h pz 1 q is reduced to tz 1 u when π h pz 1 q P TorpB hq.

-8 singular fibers of type I 1 , each of them made of a single, active component; -1 fiber made of two rational curves (intersecting transversally in two points), the first is a p´1q-curve E g (contracted by the reduction process), the second is a p´2q-curve S which reduces to a I 1 fiber, so it is active; -one multiple fiber M g of type 2I 0 .

The curve S does not depend on g, and D Γ " S, so we see that condition (AC) is violated. On the other hand, if we blow-down the p´2q-curve S onto a point, we get a singular surface on which Theorem A' applies.

The relevance of condition (AC) comes from the following lemma. We resume the context of Lemma 6.6. Lemma 6.10. Let s P Critpπ g q be such that X g s reduces to type I b , b ě 1. Let C be a branch of σ g accumulating s, of type R k p,q with q ‰ 0. Let pw n q be a sequence of points of Cz TorpB g q converging toward s. Pick an arbitrary sequence ξ n P X g wn . If A is any active component of X g s , then L 0 g pξ n q accumulates A along a non-trivial curve.

Proof. Since A is not contracted during the reduction process, we can assume that X g s is already of type I b . We now rely on the description of fibers of type I b given in § § 3.3.1 and 3.3.2. On a small disk V Ă B g containing s, we pick a local section of π g intersecting A at a smooth point of X g s and we construct the surface X g,7 V , with central fiber X g,7 s corresponding to A. Since q ‰ 0, the circle L 0 g pξ n q is not homotopic to a vanishing cycle, so its length is bounded from below by the injectivity radius of X. More precisely, we can extract a subsequence L 0 g pξ n i q that converges in X g,7

V towards a logarithmic spiral in the central fiber X g,7 s » C ˆ. Here, by a logarithmic spiral we mean a translate of a one parameter subgroup that goes from 0 to 8 in the complex multiplicative group C ˆ. The result follows.

Proof of Theorem A'. By assumption Γ satisfies (AC). We shall revisit the choice of the Halphen twists g and h from §5.2.1. First, we choose g with the required property in (AC); an extra condition will also be imposed on h (see below).

From the first lines of §6.1, we may assume Tang tt pπ g , π h q ‰ H, and we have to show that σ g admits a semi-analytic continuation to B g . Fix s P Critpπ g q and small disks V Ť V 1 centered at s, as in § 6.2. If the singular fiber X g s reduces to type I 0 then σ g is semi-analytic at s by Lemma 6.2. If it reduces to type I b , b ě 1, any branch of σ g with q " 0 is semi-analytic by Lemma 6.4. By Proposition 6.7, σ g has finitely many components near s; thus, we only need to show that any given branch of σ g with q ‰ 0 is semi-analytic.

Fix a branch γ converging to s, of type R α,β p,q for some q ‰ 0. Since Tang Γ contains D Γ , we may assume that there is an active component A Ă X g s which is not contained in any fiber of h. Now, fix an invariant, ergodic probability measure µ such that µ g evenly charges γ. We will argue as in Lemma 5.10, except that the uniform geometric estimates from §5.3 are replaced by Lemma 6.10.

The details are as follows. Let r be such that any logarithmic spiral in A contains an arc of size r. Fix ε ! δ ! r. Set k :" ˇˇTang tt pπ g , π h q X X g s ˇˇ; k ă `8 by definition of Tang tt pπ g , π h q.

(ii) Γ is a non-elementary group of automorphisms of X that contains a parabolic element g;

(iii) every parabolic element g of Γ acts on the base of its invariant fibration π g : X Ñ B g by an automorphism g B : B g Ñ B g of infinite order.

We provide an argument to complete the proof of Theorem A in that case; the strategy is the same as in Sections 4 and 5, but simpler since the dynamics is linear:

Proposition A.1. Under the above hypotheses (i), (ii), (iii), if µ is a Γ-invariant and ergodic measure, then either µ is the Haar measure on the abelian surface X, or there are finitely many subtori S j Ă X of real dimension 2, and points a j P X, j " 1, . . . , k, such that

(1) Ť j pa j `Sj q is Γ-invariant;

(2) Γ permutes transitively the subsets a j `Sj , j " 1, . . . , k;

(3) µ is supported on Ť j pa j `Sj q and on each a j `Sj , µ is given by 1 k m j where m j is the Haar measure on a j `Sj .

Here, what we call Haar measure on a j `Sj is the image of the Haar measure on S j by the translation s P S j Þ Ñ a j `s. With the results of Sections 4, this proposition concludes the proof of Theorem A.

A.1. Parabolic, affine transformations. The group Γ acts on X by affine transformations (A.1) f px, yq " A f px, yq `Sf mod pΛq where the linear part A f P GL 2 pCq preserves the lattice Λ Ă C 2 and the translation part S f is an element of C 2 {Λ. Now, pick a parabolic element g P Γ; its linear part is given by (A.2) A g " ˆ1 0 1 1 ȧfter a linear change of coordinates in C 2 . In these coordinates, the fibration π g is induced by the projection π 1 : px, yq Þ Ñ x, and a conjugation by a translation reduces g to the form (A.3) gpx, yq " px `s, y `xq mod pΛq where s has infinite order in the elliptic curve B g " C{π 1 pΛq.

Lemma A.2. If the orbits of g B : x Þ Ñ x `s are dense in B g , then g is uniquely ergodic: the unique g-invariant probability measure on X is the Haar measure.

This result is due to Furstenberg (see [24, §3.3]). Thus, in this case µ is the Haar measure on X and we are done. So in what follows, we assume that for every g P HalpΓq the orbits of the translation g B are not dense: they equidistribute along circles x `Tg , where T g is the closure of the group Zs Ă B g ; changing g into a positive iterate we may assume that this closure is isomorphic to R{Z (as a real Lie group). We let g be the quotient map C 2 {Λ Ñ B g {T g .

Lemma A.3. Every fiber of the linear projection g is a 3-dimensional g-invariant torus, and g is uniquely ergodic on almost every fiber.

To prove Lemma A.3, we think of C 2 as a real vector space and fix a basis of Λ. Then C 2 is identified with R 4 and Λ with Z 4 Ă R 4 . The eigenspace of A g for the eigenvalue 1 is defined over Z with respect to Λ " Z 4 . Moreover, A g acts trivially on the quotient space R 4 {FixpA g q. Thus adapting the basis of Z 4 to g, we may assume that

for some irrational number s 2 and some integers a, b, c, and d. The linear projection g is now given by px 1 , x 2 , x 3 , x 4 q Þ Ñ x 1 and Lemma A.3 boils down to the following statement.

Lemma A.4. If 1, s 1 and s 2 are linearly independent over Q, then g is uniquely ergodic on the level set tx 1 " s 1 u.

Proof. Let us first observe that ad ´bc ‰ 0. Indeed, otherwise the linear part of g would have a fixed point set of dimension 3, which is impossible because g is holomorphic (see Equation (A.2)). To prove unique ergodicity, we use the following criterion due to Furstenberg (see [START_REF] Furstenberg | Recurrence in ergodic theory and combinatorial number theory[END_REF]Prop. 3.10]): let h be a homeomorphism on R{Z ˆpR 2 {Z 2 q of the form px, yq Þ Ñ px `u, y `ϕpxqq, where u is irrational, then h is uniquely ergodic if and only if it is ergodic for the Haar measure. On the fiber x 1 " s 1 , our map g is of the form (A.5) px 2 , x 3 , x 4 q Þ ÝÑ px 2 `s2 , x 3 `as 1 `bx 2 , x 4 `cs 1 `dx 2 q, so we need to check that it is ergodic for the Haar measure. For this, we pick a measurable invariant subset A Ă R 3 {Z 3 , we denote by 1 A P L 2 pR 3 {Z 3 q its indicator function, and we expand it into a Fourier series 1 A px 2 , x 3 , x 4 q " ř pk, ,mqPZ 3 c k, ,m e 2iπpkx 2 ` x 3 `mx 4 q . Then (A.6) 1 A ˝gpx 2 , x 3 , x 4 q " ÿ pk, ,mqPZ 3 c k, ,m e 2iπks 2 e 2iπp a`mcqs 1 e 2iπpk` b`mdqx 2 e 2iπ x 3 e 2iπmx 4 and, from the g-invariance of 1 A and the uniqueness of the expansion, we get (A.7) c k, ,m " e 2iπks 2 e 2iπp a`mcqs 1 c k´ b´md, ,m for all pk, , mq P Z 3 . For " m " 0, the irrationality of s 2 implies that c k,0,0 " 0 unless k " 0. If b `md ‰ 0, iterating the relation |c k, ,m | " |c k´ b´md, ,m | and using the fact that Fourier coefficients decay to zero at infinity, we infer that c k, ,m " 0. Finally, if b `md " 0 and one of or m is nonzero, since ad ´bc ‰ 0 we get that a `mc ‰ 0 and (A.7) gives c k, ,m " e 2iπpks 2 `p a`mcqs 1 q c k, ,m . Since 1, s 1 , and s 2 are Q-linearly independent, we derive c k, ,m " 0. Thus, 1 A is a constant, which means that the Haar measure of A is 0 or 1.

A.2. Using distinct parabolic automorphisms. To complete the proof of Theorem A in the case of tori, one can now follow the same ideas as in Sections 4 and 5. We only need to replace the dimension dim R pµq by the minimal dimension of a real subtorus Z Ă X such that µpq Zq ą 0 for some q P X, the invariant fibration π g by the R-linear projection g , and the set R g pB g q Ă B g by (A.8)

Rpgq " ty P B g {T g ; g is not uniquely ergodic in ´1 g pyqu Ă B g {T g .

Lemma A.4 shows that Rpgq is countable.