Journées ANR Fatou

Séries lacunaires et tube de Siegel rond dans CxC.

J'exhiberai une famille d'automorphismes de CxC conjugués à une rotation sur DxC, et chaotiques en dehors de ce tube. Il s'agit d'un travail en collaboration avec Davoud Cheraghi.

A Spectral Gap for the Transfer Operator on Complex Projective Spaces.

We study the transfer (Perron-Frobenius) operator on P^k(C) induced by a generic holomorphic endomorphism and a suitable continuous weight. We prove the existence of a unique equilibrium state and we introduce various new invariant functional spaces, including a dynamical Sobolev space, on which the action of f admits a spectral gap. This is one of the most desired properties in dynamics. It allows us to obtain a list of statistical properties for the equilibrium states. Most of our results are new even in dimension 1 and in the case of constant weight function, i.e., for the operator f_*. Our construction of the invariant functional spaces uses ideas from pluripotential theory and interpolation between Banach spaces. This is a joint work with Tien-Cuong Dinh.

Analyse fonctionnelle sur les b-diviseurs et croissance des degrés.

Dans cet exposé issu d’un travail en commun avec Charles Favre, je vais expliquer comment on peut mettre plusieurs topologie de Banach sur les espace de b-diviseurs pour lesquels les applications rationnelles induisent des opérateurs bornés. Ceci permet de contrôler la croissance du degré algébrique des itérés d’applications rationnelles sous certaines hypothèses génériques.

Dynamics of isoperiodic foliations.

I will report on recent progress on the understanding of the dynamics and topology of isoperiodic foliations on moduli spaces of abelian differentials on curves.

Dynamical degrees of rational maps. (Lecture)

The dynamics of a rational map on projective space are governed by its so-called dynamical degrees. After reviewing definitions and known facts about dynamical degrees, I will focus on maps in dimensions two and three and describe in detail some recent examples in which the first dynamical degree turns out to be a transcendental number.

Dynamics of groups of automorphisms of compact complex surfaces. (Lecture)

In this series of lectures we will study the dynamics of “large” groups of algebraic automorphisms of projective surfaces. Depending on time, the topics covered should include:
• the classification of invariant measures and orbit closures
• finiteness properties of the set of finite orbits
• uniform hyperbolicity and applications

On the complement of minimal sets of foliations.

The complement of minimal sets of holomorphic foliations usually display interesting complex analytic properties. This is the case of Grauert’s famous example, which provides pseudoconvex domains in complex tori which are not Stein. In this talk we present the following result: the complement of the limit set of certain Riccati foliations is strongly pseudoconvex. For the proof we construct a strictly psh exhaustion function near the limit set, by using the dynamics of the foliation and the transverse geometry of the limit set. This is a joint work with B. Deroin and V. Kleptsyn.

Rigidité non-Archimédienne.

Nous expliquerons comment les techniques non-Archimédiennes permettent d'analyser les familles de fractions rationnelles dont les multiplicateurs des orbites périodiques restent bornés.

Bifurcation loci of families of finite type meromorphic maps.

In this talk we will discuss parameter spaces of natural families of transcendental meromorphic maps of finite type, parametrized over a complex manifold. In this setting, a new type of bifurcation arises for which a periodic cycle can disappear to infinity along a parameter curve. We shall relate these type of bifurcation parameters, with those for which an asymptotic value is a prepole, and use these relations to study stability of Julia sets (J-stability), concluding that J-stable parameters form an open and dense subset of the parameter space. All our theorems hold for general finite type maps in the sense of Epstein, satisfying certain conditions. This is joint work with Matthieu Astorg and Anna Miriam Benini.

Variations of canonical height and equidistribution: a complex analytic approach.

An important objet in the study of the dynamics of an endomorphism of a projective space defined over a number field is its canonical height. It essentially measures how far a point with algebraic coordinates is from being pre-periodic. When studying families of such endomorphisms which are defined over number fields, one may wonder how those objects vary with the parameter. In this talk, I will motivate the study of these objects, as well as the problem we are investigating. We will make connections with the notions of bifurcation measure and, if time allows, we will discuss applications to equidistribution and sparsity problems in parameter spaces.

A flower theorem in dimension two.

The local dynamics of a tangent to the identity biholomorphism in dimension one is described by Leau-Fatou flower theorem. We present a two-dimensional version of this theorem, valid when the fixed point is a non-degenerate singular point. This is a joint work with Rudy Rosas.

Dynamics of Koch postcritically finite endomorphisms.

Most examples of postcritically finite endomorphisms of projective spaces of dimension at least 2 can be recovered from a construction due to Koch, which is related to Thurston's topological characterization of postcritically finite rational maps. In this talk, I will study the dynamics of these endomorphisms on their postcritical locus and examine the eigenvalues associated with their periodic points.

Random walks on SL_2(C) via holomorphic dynamics.

Given a sequence of random i.i.d. 2 by 2 complex matrices, it is a classical problem to study the statistical properties of their product. This theory dates back to fundamental works of Furstenberg, Kesten, etc. and is still an active research topic. In this talk, I intend to show how methods from complex analysis and analogies with holomorphic dynamics offer a new point of view to this problem. This is used to obtain several new limit theorems for these random processes, often in their optimal version. This is based on joint works with T.-C. Dinh and H. Wu.

Approximation of summable rational maps.

In this talk, I will present how rational maps which satisfy a summability condition can be approximated by hyperbolic rational maps. I will sketch the proof of this result which is based on an adaptation of the work of Gao and Shen into the complex case. This work is part of my thesis under the supervision of T. Gauthier and G. Vigny.

Courants de bifurcations pour les familles de représentations de groupes en rang supérieur.

Les représentations linéaires agissant sur les espaces projectifs sont des systèmes dynamiques holomorphes qui exhibent une grande variété de comportements. Nous introduirons la notion de stabilité proximal, une généralisation de la stabilité au sens de Sullivan, qui mesure une certaine forme de stabilité dynamique de l’action d’une famille holomorphe de représentations et nous expliquerons comment cette propriété est détectée par un un courant de bifurcation sur l’espace des paramètres de la famille. Ce courant de bifurcation mesure la pluriharmonicité du plus grand exposant de Lyapunov de la famille de représentation, associé à une marche aléatoire.

Germs tangent to the identity in higher dimension. (Lecture)

In this series of lectures, we will study the dynamics of germs tangents to the identity at a fixed point. After briefly reviewing the classical one-dimensional results, I will focus on the dynamics in dimension two. Depending on time, the topics covered should include: parabolic curves, parabolic domains, spiralling domains, wandering domains, parabolic implosion.

Birational properties of tangent to the identity germs in dimension 3.

I will present a family of tangent-to-the-identity germs in dimension 3 associated to isolated fixed points, for which the lift to any modification (adapted to the dynamics) has only degenerate characteristic directions (in sharp contrast with what happens in dimension 2). For such germs, I will also discuss the existence of formal invariant curves and their associated parabolic manifolds, following recent techniques developed by López-Hernanz, Raissy, Ribón, Sanz-Sánchez, Vivas.
This is a joint work with Samuele Mongodi.

Dynamics of horizontal-like maps in higher dimension.

The set of horizontal-like maps is a very large family of dynamical systems which includes generalized Hénon maps in C² and regular automorphisms of C^k in the sense of Sibony. I will review some dynamical properties of the Green currents, equilibrium measures, dynamical degrees, and entropies of horizontal-like maps. I will also present some recent progress in this research direction. This talk is based on my joint works with F. Bianchi, V.-A. Nguyen, K. Rakhimov and N. Sibony.

Northcott property in the quadratic family.

I will discuss two notions of stability for algebraic sets A of dimension 1 in C² in the quadratic family (f:(l,z)->(l, z²+l)):
• global bound of the degree of the iterates deg(fⁿ(A))=O(1).
• Local stability in the sense that the sequence of iterates of the marked point corresponding to A (say A is a graph over the l variable) is locally equicontinuous.

I will show that these notions are in fact equivalent and that A is stable if and only if it is preperiodic (j.w. with Gauthier).

On Hausdorff dimension of polynomial not totally disconnected Julia sets.

I will present recent results obtained in a joint work with Feliks Przytycki. We prove that for every polynomial of one complex variable of degree at least 2 and Julia set not being totally disconnected nor a circle, nor interval, Hausdorff dimension of this Julia set is larger than 1. Till now this was known only in the connected Julia set case. We give also an example of a polynomial with non-connected but not totally disconnected Julia set and such that all its components comprising of more than single points are analytic arcs, thus resolving a question by Christopher Bishop, who asked whether every such component must have Hausdorff dimension larger than 1. In my talk, I will also discuss some related questions and conjectures.