# Abstracts

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**MINICOURSES**

**Pierre BERGER (CNRS - Paris 13)**

*Abundance of one dimensional non-uniformly hyperbolic attractors for surface endomorphisms*

We present a (new) proof of the existence of a non uniformly hyperbolic attractor for a positive set of parameters *a* in the family of endomorphisms:

*(x,y) --> (x ^{2}+a+y,0)+B(x,y),*

where *B* is any fixed *C ^{2}* small function. For

*B=0*, this is the Jackoson theorem. For

*B=b (0,x)*, we get the Benedicts-Carleson (B-C) theorem for the Hénon map.

The proof is done thanks to analytical and probabilistic tools of (B-C) in the geometric and combinatorial formalism of Yoccoz puzzles generalized in a very algebraic way (pseudo-semi-group). These theorems are notably generalized to the

*C*-case and to the endomorphisms. The theorem is an answer to a question of Pesin-Yurchenko reaction-diffusion PDEs in applied mathematics.

^{2}A short summary on this proof is avalaible at:

http://www.math.sunysb.edu/~berger/bergermfo.pdf

The lastest version of this proof is:

http://www.math.sunysb.edu/~berger/henonparabolic.pdf

**David DAMANIK (Rice)
Non-uniformly hyperbolic Schrödinger equations
**

We will discuss Schrödinger operators with dynamically defined potentials and the associated stationary Schrödinger equation. This equation can be rewritten in a straightforward way as the iteration of an SL(2,R)-valued cocycle over the base dynamics. The spectral properties of the Schrödinger operator in question are closely related to the dynamics of the cocycle. In particular, a number of consequences can be drawn in cases where the cocycle is non-uniformly hyperbolic. This minicourse will explain this connection, present ways of establishing non-uniform hyperbolicity of the cocycle, and discuss signatures of non-uniform hyperbolicity on a spectral level.

Ian MELBOURNE (Surrey)

*Mixing and Decay of Correlations for Nonuniformly Hyperbolic flows*

This minicourse will consist primarily of the following components:

(i) A review of the state-of-the-art on mixing and rates of mixing for uniformly and non uniformly hyperbolic flows.

(ii) Proofs of the following results for uniformly hyperbolic flows: Mixing is typical (Bowen); Superpolynomial decay is typical (Dolgopyat); These properties are typically stable, *i.e.*, they hold for open and dense subsets of such flows (Field, Melbourne & Török).

(iii) Extensions to nonuniformly hyperbolic flows, including Lorentz gases, Lorentz gases in domains with cusps, suspended Hénon-like attractors, (geometric) Lorenz attractors.

(iv) Decay of correlations for slowly mixing flows including infinite horizon Lorentz gases.

The minicourse will focus on methods that have been shown to apply for large classes of systems. Exponential decay of correlations (probably the correct result in many situations) has been established only in very special cases, and will be de-emphasized for this reason.

**TALKS**

**Henk BRUIN**

*Monotonicity of topological entropy in polynomial families of maps*

**Topological entropy is a measure of complexity of a dynamical system. For chaotic dynamical systems as basic as the quadratic family**

f

_{a}(x) = ax(1-x), it has been observed that period doubling bifurcations are always occurring in the same way as the parameter a increases. Monotone dependence of topological entropy on the parameter is a corollary of this.

Since the first proof by Milnor & Thurston, several other proofs of monotonicity of entropy have been given, but all of them rely on complex analysis. In 2000, Milnor & Tresser proved monotonicity for the two-parameter cubic family. Now, monotonicity means that the level sets of constant entropy are connected subsets of parameter space. In this talk I want to discuss joint work with Sebastian van Strien in which we prove the general case, i.e., monotonicity for polynomial families of any degree. Still, part of the proof relies on methods from complex analysis.

**Roman HRIC**

*Almost totally disconnected minimal systems*

In this talk I start with a brief survey of old and recent results on topological structure of minimal sets in dynamical systems. Then I present new results from a joint paper by Balibrea, Downarowicz, Hric, Snoha and Špitalský, of the same title as the talk.

We construct a new rich class of minimal systems - almost totally disconnected minimal systems. A topological space is said to be almost totally disconnected if the set of its degenerate components is dense. We prove that an almost totally disconnected compact metric space admits a minimal map if and only if either it is a finite set or it has no isolated point. As a consequence we obtain a topological characterization of minimal sets on dendrites and local dendrites. We also prove that any infinite compact almost totally disconnected space with no isolated point admits a minimal map with arbitrary entropy.

Renaud LEPLAIDEUR

*CLT for Hausdorff dimension of Gibbs measures*

For skew (uniformly) expanding maps we prove a Central Limit Theorem for the Hausdorff dimension of Gibbs measures. We would like to focus on the point that, although we need the uniformly expanding settings to define the Gibbs measures, the proof essentially uses non uniformly expanding arguments *a la Pesin*. We thus expect that the same method could be applied to non-uniformly expanding maps... as soon as Gibbs measures exist.

This is a joint work with Benoît Saussol.

**Krerley OLIVEIRA
**

**Periodic orbits, Lyapunov Exponents and Recurrence**

Periodic orbits are one main actor in dynamical systems. Despite the fact that in some setting they are extremely difficult to obtain, under a "sufficient chaotic" situation there are plenty of them. How they are distributed plays a important role in the study of dynamical systems.

In this talk we prove a general version of the well-known "Katok´s Closing Lemma". Given a (ergodic) invariant measure for C1 a dynamical system with only positive Lyapunov exponents, we are able to show that almost every point is shadowed by a periodic orbit with period that growth sublinearly with the size of the piece of orbit that you wanna shadow. We discuss some interesting applications on recurrence estimates and approximations by periodic measures.

Jimmy SANTAMARIA

Jimmy SANTAMARIA

**Non-zero Lyapunov exponents of linear cocycles over partially****hyperbolic diffeomorphisms**

We will discuss results of Avila-Santamaria-Viana about holonomy invariance as measurable and continuous rigidity properties of satured sections (of certain bundles) with respect to a partially hyperbolic diffeomorphism.

We will rephrase these results as necessary conditions for a fiber bunched linear cocycle, over a center bunched accessible partially hyperbolic diffeomorphism, to have all its Lyapunov exponents equal to zero and see how this could be broken genererically. Also survey an application given by Wilkinson in establishing measurable rigidity of solutions to the cohomological equation over center-bunched diffeomorphisms. We will also state Avila-Bochi-Wilkinson generalization of these results for partially hyperbolic maps not necessarily center bunched.

**
Wenxiang SUN
** In approximation direction, we prove that invariant measures in a nonuniformly hyperbolic system can be approximated by periodic measures, and the Lyapunov exponents, mean angles and independece numbers of an ergodic hyperbolic measure can be approximated by that of periodic measures.

*Approximation properties and deviation bounds for nonuniformly hyperbolic systems*

In large deviation direction, we bound the periodic measures far from an ergodic hyperbolic measure, and bound periodic measures whose Lyapunov exponents, mean angle and independece number are far that of an ergodic hyperbolic measure.

Andrew TÖRÖK

Andrew TÖRÖK

*Transitivity of Euclidean-type extensions of hyperbolic systems*

Let f, defined from X to X, be the restriction to a hyperbolic basic set of a smooth diffeomorphism. For r>0, we show that in the class of C

^{r}-cocycles with fiber the special Euclidean group SE(n) those that are transitive form a residual set (countable intersection of open dense sets). This result is new for n>=3 odd.

More generally, we consider Euclidean-type groups G ∝ R

^{n}where G is a compact connected Lie group acting linearly on R

^{n}. When FixG = {0}, it is again the case that the transitive cocycles are residual. When FixG ≠ {0}, the same result holds on the subset of cocycles that avoid an obvious and explicit obstruction to transitivity.

This is joint work with Ian Melbourne and Viorel Niţicǎ

**Francisco VALENZUELA**

*Critical point for generalized Hénon maps*

Generalized Hénon maps are the polynomial automorphisms of C2 that represent the first step for a global understanding of holomorphic dynamics in higher dimension. As in the one dimensional context (polynomials on C), it can be dened the Julia set, which is the set that concentrate the interesting dynamical behavior of these type of systems.

The main goal is to understand the dynamical obstruction for domination in the Julia set. This allow us to introduce a notion of critical point (and critical set) for polynomial automor-phisms that capture many of the dynamical properties of their one-dimensional counterpart.