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Under a natural assumption on the dynamical degrees, we prove that the Green currents associated to any H\'enon-like map in any dimension have H\"older continuous super-potentials, i.e., give H\"older continuous linear functionals on suitable spaces of forms and currents. As a consequence, the unique measure of maximal entropy is the Monge-Amp\`ere of a H\"older continuous plurisubharmonic function and has strictly positive Hausdorff dimension. Under the same assumptions, we also prove that the Green currents are woven. When they are of bidegree $(1,1)$, they are laminar. In particular, our results generalize results known until now only in algebraic settings, or in dimension 2.

Let $X$ be a compact Riemann surface and $\mathcal L$ be a positive line bundle on it. We study the conditional zero expectation of all the holomorphic sections of $\mathcal L^n$ which do not vanish on $D$ for some fixed open subset $D$ of $X$. We prove that as $n$ tends to infinity, the zeros of these sections are equidistributed outside $D$ with respect to a probability measure $\nu$. This gives rise to a surprising forbidden set., Comment: fisrt draft

We propose a set of questions on the dynamics of H\'enon maps from the real, complex, algebraic and arithmetic points of view., Comment: 34 pages, 4 figures

Let $f\colon X\to X$ be a dominant meromorphic self-map of a compact complex variety $X$ in the Fujiki class $\mathcal{C}$. If the topological degree of $f$ is strictly larger than the other dynamical degrees of $f$, we show that the number of isolated $f$-periodic points grows exponentially fast similarly to the topological degrees of the iterates of $f$; in particular, we give a positive answer to a conjecture of Shou-Wu Zhang. In the general case, we show that the exponential growth of the number of isolated $f$-periodic points is at most the algebraic entropy of $f$., Comment: 22 pages, comments welcome!

Let $X$ be a compact K\"ahler manifold. We study subgroups $G \le \mathrm{Aut}(X)$ of biholomorphic automorphisms of zero entropy when $\mathrm{Aut}^0(X)$ is compact (e.g. when $\mathrm{Aut}^0(X)$ is trivial). We show that the virtual derived length $\ell_{\mathrm{vir}}(G)$ of $G$ satisfies $\ell_{\mathrm{vir}}(G) \le \dim X -\kappa(X)$, where $\kappa(X)$ is the Kodaira dimension of $X$. Modulo the main conjecture of our previous work concerning the essential nilpotency class, we obtain the same upper bound $c_{\mathrm{vir}}(G) \le \dim X -\kappa(X)$ for the virtual nilpotency class $c_{\mathrm{vir}}(G)$, together with a geometric description of the $G$-action on $X$ when the equality holds., Comment: This paper contains results from an earlier version of arXiv:1810.04827. They are completely revised with new proofs. Comments are welcome

We prove that, for every invertible horizontal-like map (i.e., H{\'e}non-like map) in any dimension, the sequence of the dynamical degrees is increasing until that of maximal value, which is the main dynamical degree, and decreasing after that. Similarly, for polynomial-like maps in any dimension, the sequence of dynamical degrees is increasing until the last one, which is the topological degree. This is the first time that such a property is proved outside of the algebraic setting. Our proof is based on the construction of a suitable deformation for positive closed currents, which relies on tools from pluripotential theory and the solution of the $d$, $\bar \partial$, and $dd^c$ equations on convex domains.

We consider the unique measure of maximal entropy of an automorphism of a compact K{\"a}hler manifold with simple action on cohomology. We show that it is exponentially mixing of all orders with respect to H{\"o}lder observables. It follows that the Central Limit Theorem (CLT) holds for these observables. In particular, our result applies to all automorphisms of compact K{\"a}hler surfaces with positive entropy.

We show that the measure of maximal entropy of every complex H\'enon map is exponentially mixing of all orders for H\''older observables. As a consequence, the Central Limit Theorem holds for all H\''older observables.

Let $f$ be a meromorphic correspondence on a compact K\"ahler manifold $X$ of dimension $k$. Assume that its topological degree is larger than the dynamical degree of order $k-1$. We obtain a quantitative regularity of the equilibrium measure of $f$ in terms of its super-potentials.

The aim of this paper is twofold. First of all, we confirm a few basic criteria of the finiteness of real forms of a given smooth complex projective variety, in terms of the Galois cohomology set of the discrete part of the automorphism group, the cone conjecture and the topological entropy. We then apply them to show that a smooth complex projective surface has at most finitely many non-isomorphic real forms unless it is either rational or a non-minimal surface birational to either a K3 surface or an Enriques surface. In the second part of the paper, we construct an Enriques surface whose blow-up at one point admits infinitely many non-isomorphic real forms. This answers a question of Kondo to us and also shows the three exceptional cases really occur., Comment: v2: the structure of the paper has been changed to improve the exposition; the blow-up at one point of an Enriques surface with infinitely many real forms that we construct is now proved to have non-finitely generated automorphism group

We establish the existence of a spectral gap for the transfer operator induced on $\mathbb P^k = \mathbb P^k (\mathbb C)$ by a generic holomorphic endomorphism and a suitable continuous weight and its perturbations on various functional spaces, which is new even in dimension one. The main issue to overcome is the rigidity of the complex objects, since the transfer operator is a non-holomorphic perturbation of the operator $f_*$. The system is moreover non-uniformly hyperbolic and one may have critical points on the Julia set. The construction of our norm requires the introduction and study of several intermediate new norms, and a careful combination of ideas from pluripotential and interpolation theory. As far as we know, this is the first time that pluripotential methods have been applied to solve a mixed real-complex problem. Thanks to the spectral gap, we establish an exponential speed of convergence for the equidistribution of the backward orbits of points towards the conformal measure. Moreover, we obtain a full list of statistical properties for the equilibrium states: exponential mixing, CLT, Berry-Esseen theorem, local CLT, ASIP, LIL, LDP, almost sure CLT. Many of these properties are new even in dimension one, some even in the case of zero weight function (i.e., for the measure of maximal entropy)., Comment: Part 2 of the original submission arxiv:2007.04595v3

Let $\mu$ be a probability measure on $\text{GL}_d(\mathbb R)$ and denote by $S_n:= g_n \cdots g_1$ the associated random matrix product, where $g_j$'s are i.i.d.'s with law $\mu$. We study statistical properties of random variables of the form $$\sigma(S_n,x) + u(S_n x),$$ where $x \in \mathbb P^{d-1}$, $\sigma$ is the norm cocycle and $u$ belongs to a class of admissible functions on $\mathbb P^{d-1}$ with values in $\mathbb R \cup \{\pm \infty\}$. Assuming that $\mu$ has a finite exponential moment and generates a proximal and strongly irreducible semigroup, we obtain optimal Berry-Esseen bounds and the Local Limit Theorem for such variables using a large class of observables on $\mathbb R$ and H\"older continuous target functions on $\mathbb P^{d-1}$. As particular cases, we obtain new limit theorems for $\sigma(S_n,x)$ and for the coefficients of $S_n$., Comment: Major modifications from v1: more general results with unified proofs allowing admissible functions u. Results for the norm cocycle and the coefficients are now particular cases. Proof of the Local Limit Theorem included. arXiv admin note: text overlap with arXiv:2110.09032

Let $\mu$ be a probability measure on $\text{GL}_d(\mathbb{R})$ and denote by $S_n:= g_n \cdots g_1$ the associated random matrix product, where $g_j$ are i.i.d. with law $\mu$. Under the assumptions that $\mu$ has a finite exponential moment and generates a proximal and strongly irreducible semigroup, we prove a Berry-Esseen bound with the optimal rate $O(1/\sqrt n)$ and a general Local Limit Theorem for the coefficients of $S_n$., Comment: 27 pages. arXiv admin note: text overlap with arXiv:2106.04019

Let $\nu$ be the Furstenberg measure associated with a non-elementary probability measure $\mu$ on SL_2(R). We show that, when $\mu$ has a finite second moment, the Fourier coefficients of $\nu$ tend to zero at infinity. In other words, $\nu$ is a Rajchman measure. This improves a recent result of Jialun Li., Comment: 50 pages, v1->v2: only minor modifications

The classical Marcinkiewicz theorem states that if an entire characteristic function $\Psi$ of a non-degenerate real-valued random variable $X$ is of the form $\exp(P(u))$ for some polynomial $P$, then $X$ has to be a Gaussian. In this work, we obtain a broad, quantitative extension of this framework in several directions, establish central limit theorems (CLTs) with explicit rates of convergence, and demonstrate Gaussian fluctuations in continuous spin systems and general classes of point processes. In particular, we obtain quantitative decay estimates on the Kolmogorov-Smirnov distance between $X$ and a Gaussian under the condition that $\Psi$ does not vanish only on a bounded disk. This leads to quantitative CLTs applicable to very general and possibly strongly dependent random systems. In spite of the general applicability, our rates for the CLT match the classic Berry-Esseen bounds for independent sums up to a log factor. We implement this programme for two important classes of models in probability and statistical physics. First, we extend to the setting of continuous spins a popular paradigm for obtaining CLTs for discrete spin systems that is based on the theory of Lee-Yang zeros, focussing in particular on the XY model, Heisenberg ferromagnets and generalised Ising models. Secondly, we establish Gaussian fluctuations for linear statistics of so-called $\alpha$-determinantal processes for $\alpha \in \mathbb{R}$ (including the usual determinantal, Poisson and permanental processes) under very general conditions, including in particular higher dimensional settings where structural alternatives such as random matrix techniques are not available. Our applications demonstrate the significance of having to control the characteristic function only on a (small) disk, and lead to CLTs which, to the best of our knowledge, are not known in generality.

We construct a smooth complex projective rational surface with infinitely many mutually non-isomorphic real forms. This gives the first definite answer to a long standing open question if a smooth complex projective rational surface has only finitely many non-isomorphic real forms or not., Comment: final form to appear in J. Reine Angew. Math

We obtain various new limit theorems for random walks on SL_2(C) under low moment conditions. For non-elementary measures with a finite second moment, we prove a Local Limit Theorem for the norm cocycle, yielding the optimal version of a theorem of E. Le Page. For measures with a finite third moment, we obtain the Local Limit Theorem for the matrix coefficients, improving a recent result of Grama-Quint-Xiao and the authors, and Berry-Esseen bounds with optimal rate $O(1 / \sqrt n)$ for the norm cocycle and the matrix coefficients. The main tool is a detailed study of the spectral properties of the Markov operator and its purely imaginary perturbations acting on different function spaces. We introduce, in particular, a new function space derived from the Sobolev space $W^{1,2}$ that provides uniform estimates., Comment: 64 pages. Revised version. To appear in Probab. Theory Relat. Fields

We study local positive harmonic currents directed by a foliation by Riemann surfaces near a hyperbolic singularity which have no mass on the separatrices. A theorem of Nguy\^en says that the Lelong number of such a current at the singular point vanishes. We prove that this property is sharp: one cannot have any better mass estimate for this current near the singularity.

Let $X$ be a compact K\"ahler manifold of dimension $n$ and $\omega$ a K\"ahler form on $X$. We consider the complex Monge-Amp\`ere equation $(dd^c u+\omega)^n=\mu$, where $\mu$ is a given positive measure on $X$ of suitable mass and $u$ is an $\omega$-plurisubharmonic function. We show that the equation admits a H\"older continuous solution {\it if and only if} the measure $\mu$, seen as a functional on a complex Sobolev space $W^*(X)$, is H\"older continuous. A similar result is also obtained for the complex Monge-Amp\`ere equations on domains of $\mathbb{C}^n$., Comment: 16 pages. Final version, to appear in Bulletin of the London Mathematical Society

We develop a new method, based on pluripotential theory, to study the transfer (Perron-Frobenius) operator induced on $\mathbb P^k = \mathbb P^k (\mathbb C)$ by a holomorphic endomorphism and a suitable continuous weight. This method allows us to prove the existence and uniqueness of the equilibrium state and conformal measure for very general weights (due to Denker-Przytycki-Urba\'nski in dimension 1 and Urba\'nski-Zdunik in higher dimensions, both in the case of H\''older continuous weights). We establish a number of properties of the equilibrium states, including mixing, K-mixing, mixing of all orders, and an equidistribution of repelling periodic points. Our analytic method replaces all distortion estimates on inverse branches with a unique, global, estimate on dynamical currents, and allows us to reduce the dynamical questions to comparisons between currents and their potentials., Comment: Part 1 of the previous version. Part 2 to be submitted separately

Our aim is to give a version of the Moser-Trudinger inequality in the setting of complex geometry. As a very particular case, our result already gives a new Moser-Trudinger inequality for functions in the Sobolev space $W^{1,2}$ of a domain in $R^2$. We also deduce a new necessary condition for the complex Monge-Ampere equation for a given measure on a compact Kahler manifold to admit a Holder continuous solution., Comment: Final version, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)

We show, among other things, that for each integer $n \ge 3$, there is a smooth complex projective rational variety of dimension $n$, with discrete non-finitely generated automorphism group and with infinitely many mutually non-isomorphic real forms. Our result is inspired by the work of Lesieutre and the work of Dinh and Oguiso., Comment: Title has been changed following referee. the final form to appear in Math. Ann

Working within the polynomial quadratic family, we introduce a new point of view on bifurcations which naturally allows to see the seat of bifurcations as the projection of a Julia set of a complex dynamical system in dimension three. We expect our approach to be extendable to other holomorphic families of dynamical systems.

We study random products of matrices in SL_2(C) from the point of view of holomorphic dynamics. For non-elementary measures with finite first moment we obtain the exponential convergence towards the stationary measure in Sobolev norm. As a consequence we obtain the exponentially fast equidistribution of forward images of points towards the stationary measure. We also give a new proof of the Central Limit Theorem for the norm cocycle under a second moment condition, originally due to Benoist-Quint, and obtain some general regularity results for stationary measures., Comment: 27 pages

We recover results by Ullmo-Yafaev and Peterzil-Starchenko on the closure of the image of an algebraic variety in a compact complex torus. Our approach uses directed closed currents and allows us to extend the result for dimension 1 flows to the setting of commutative complex Lie groups which are not necessarily compact. A version of the classical Ax-Lindemann-Weierstrass theorem for commutative complex Lie groups is also given., Comment: 31 pages

Let \Fc be a holomorphic foliation by Riemann surfaces on a compact K\"ahler surface X. Assume it is generic in the sense that all the singularities are hyperbolic and that the foliation admits no directed positive closed (1,1)-current. Then there exists a unique (up to a multiplicative constant) positive \ddc-closed (1,1)-current directed by \Fc. This is a very strong ergodic property of \Fc. Our proof uses an extension of the theory of densities to a class of non-\ddc-closed currents. A complete description of the cone of directed positive \ddc-closed (1,1)-currents is also given when \Fc admits directed positive closed currents., Comment: Main results improved, proofs simplified, presentation changed. 50 pages

We study zero entropy automorphisms of a compact K\"ahler manifold $X$. Our goal is to bring to light some new structures of the action on the cohomology of $X$, in terms of the so-called dynamical filtrations on $H^{1,1}(X, {\mathbb R})$. Based on these filtrations, we obtain the first general upper bound on the polynomial growth of the iterations $(g^m)^* \, {\circlearrowleft} \, H^2(X, {\mathbb C})$ where $g$ is a zero entropy automorphism, in terms of ${\rm dim} \, X$ only. We also give an upper bound for the (essential) derived length $\ell_{\rm ess}(G, X)$ for every zero entropy subgroup $G$, again in terms of the dimension of $X$ only. We propose a conjectural upper bound for the essential nilpotency class $c_{\rm ess}(G,X)$ of a zero entropy subgroup $G$. Finally, we construct examples showing that our upper bound of the polynomial growth (as well as the conjectural upper bound of $c_{\rm ess}(G,X)$) are optimal., Comment: Geometric and Functional Analysis (2022), to appear; Title shortened a bit; Part of the content is moved to another paper in progress

We study the dynamics of holomorphic correspondences $f$ on a compact Riemann surface $X$ in the case, so far not well understood, where $f$ and $f^{-1}$ have the same topological degree. Under a mild and necessary condition that we call non weak modularity, $f$ admits two canonical probability measures $\mu^+$ and $\mu^-$ which are invariant by $f^*$ and $f_*$ respectively. If the critical values of $f$ (resp. $f^{-1}$) are not periodic, the backward (resp. forward) orbit of any point $a \in X$ equidistributes towards $\mu^+$ (resp. $\mu^-$), uniformly in $a$ and exponentially fast.

We present a list of open questions in the theory of holomorphic foliations, possibly with singularities. Some problems have been around for a while, others are very accessible., Comment: 10 pages

We show that there is a smooth complex projective variety, of any dimension greater than or equal to two, whose automorphism group is discrete and not finitely generated. Moreover, this variety admits infinitely many real forms which are mutually non-isomorphic over the real number field. Our result is inspired by the work of Lesieutre and answers questions by Dolgachev, Esnault and Lesieutre., Comment: final form to appear in Duke Math. J

We prove that if a positive closed current is bounded by another one with bounded, continuous or Hoelder continuous super-potentials, then it inherits the same property. There are two different methods to define wedge-products of positive closed currents of arbitrary bi-degree on compact Kaehler manifolds using super-potentials and densities. When the first method applies, we show that the second method also applies and gives the same result. As an application, we obtain a sharp upper bound for the number of isolated periodic points of holomorphic maps on compact Kaehler manifolds whose actions on cohomology are simple. A similar result still holds for a large class of holomorphic correspondences., Comment: 28 pages; dedicated to Professor Le Tuan Hoa on the occasion of his sixtieth birthday

Let -Delta+V be the Schrodinger operator acting on L^2(R^d,C) with d odd larger than 2. Here V is a bounded real- or complex-valued function vanishing outside the closed ball of center 0 and radius a. If V belongs to the class of potentials introduced by Christiansen, we show that when r goes to infinity, the resonances of -Delta+V, scaled down by the factor r, are asymptotically distributed, with respect to an explicit probability distribution on the closed lower unit half-disc of the complex plane. The rate of convergence is also considered for subclasses of potentials., Comment: 19 pages

We establish a large deviation theorem for the empirical spectral distribution of random covariance matrices whose entries are independent random variables with mean 0, variance 1 and having controlled forth moments. Some new properties of Laguerre polynomials are also given., Comment: 21 pages

We give abstract versions of the large deviation theorem for the distribution of zeros of polynomials and apply them to the characteristic polynomials of Hermitian random matrices. We obtain new estimates related to the local semi-circular law for the empirical spectral distribution of these matrices when the 4th moments of their entries are controlled., Comment: 19 pages

Equidistribution of the orbits of points, subvarieties or of periodic points in complex dynamics is a fundamental problem. It is often related to strong ergodic properties of the dynamical system and to a deep understanding of analytic cycles, or more generally positive closed currents, of arbitrary dimension and degree. The later topic includes the study of the potentials and super-potentials of positive closed currents, their intersection with or without dimension excess. In this paper, we will survey some results and tools developed during the last two decades. Related concepts, new techniques and open problems will be presented., Comment: Revised version

Let L be a positive line bundle over a projective complex manifold X. Consider the space of holomorphic sections of the tensor power of order p of L. The determinant of a basis of this space, together with some given probability measure on a weighted compact set in X, induces naturally a beta-ensemble, i.e., a random point process on the compact set. Physically, this general setting corresponds to a gas of free fermions in X and may admit some random matrix models. The empirical measures, associated with such beta-ensembles, converge almost surely to an equilibrium measure when p goes to infinity. We establish a large deviations principle (LDP) with an effective speed of convergence for these empirical measures. Our study covers the case of some beta-ensembles on a compact subset of a real sphere or of a real Euclidean space., Comment: 19 pages. arXiv admin note: text overlap with arXiv:1505.08050

Let L be a positive line bundle on a projective complex manifold. We study the asymptotic behavior of Bergman kernels associated with the tensor powers L^p of L as p tends to infinity. The emphasis is the dependence of the uniform estimates on the positivity of the Chern form of the metric on L. This situation appears naturally when we approximate a semi-positive singular metric by smooth positively curved metrics., Comment: 15 pages

We show that any dominant meromorphic self-map f of a compact Kaehler manifold X is an Artin-Mazur map. More precisely, if P_n(f) is the number of its isolated periodic points of period n (counted with multiplicity), then P_n(f) grows at most exponentially fast with respect to n and the exponential rate is at most equal to the algebraic entropy of f. Further estimates are given when X is a surface. Among the techniques introduced in this paper, the h-dimension of the density between two arbitrary positive closed currents on a compact Kaehler surface is obtained., Comment: 18 pages