## MAIN CONFERENCE

Hyperbolicity and dimension

Dates: 02-06 December 2013 at CIRM (Marseille, France)

## ABSTRACTS ANS POSTERS

**Vaughn Climenhaga: Non-uniform specification, thermodynamic formalism, and towers**

*The specification property and its variants are important for many results in dimension theory of hyperbolic systems, either directly or via the thermodynamic formalism. I will describe joint work with Dan Thompson, in which we introduce a non-uniform specification property that yields strong thermodynamic results. I will also discuss the connection between this property and tower constructions that lead to further statistical properties.*

**Lorenzo J. Diaz: Robust existence of non-hyperbolic ergodic measures with positive entropy**

*We present a method to construct ergodic non-hyperbolic measures with positive entropy. This method involves a so-called flip-flop configuration that is derived from the existence of a blender.Using this method we prove that if $\mathcal{U}$ is a $C^1$-open set of diffeomorphism where a homoclinic class contains a pair of saddles of different indices then open and densely in $\mathcal{U}$ the diffeomorphisms have ergodic non-hyperbolic measures with positive entropy supported in the class.*

**Kenneth Falconer: A Survey of Self-affine Sets and Measures**

*The talk will survey the problem of finding or estimating the Hausdorff and box dimensions of self-affine fractals that arise from hyperbolic iterated functions systems. Various results and open questions will be discussed, including recent work on almost self-affine sets and measures.*

**Aihua Fan: Almost everywhere convergence of ergodic series on hyperbolic systems**

*We consider ergodic series of the form $$ \sum_{n=1}^\infty a_n f (T^n x) $$ where $f$ is of zero mean with respect to an invariant measure $\mu$. We find conditions on the system $T$ and on the function $f$ such that the series converges $\mu$-almost everywhere provided $\sum_{n=1}^\infty |a_n|^2 <\infty$.For general orthogonal or quasi-orthogonal series, Rademacher-Menshov's sufficient condition is $\sum_{n=1}^\infty \log^2 n |a_n|^2 <\infty$.For hyperbolic systems, the series is quasi-orthogonal, which is a consequenceof mixing property. Our study involves the 3-fold mixing. Assume $\mu$ is a Gibbs measure. Let $L$ be the normalized transfer operator acting on a function space larger than the Holder space. One set of conditions we find is : $f$ is in the function space and $\sum\|L^n\|< \infty$.*

**José Ferreira Alves: SRB measures for partially hyperbolic systems whose central direction is weakly expanding**

*We consider partially hyperbolic $C^{1+}$ diffeomorphisms of compact Riemannian manifolds of arbitrary dimension which admit a partially hyperbolic tangent bundle decomposition $E^s\oplus E^{cu}$. Assuming the existence of a set of positive Lebesgue measure on which $f$ satisfies a weak nonuniform expansivity assumption in the centre~unstable direction, we prove that there exists at most a finite number of transitive attractors each of which supports an SRB measure. As part of our argument, we prove that each attractor admits a Gibbs-Markov-Young geometric structure with integrable return times. (Joint work with with C. Dias, S. Luzzatto and V. Pinheiro.)*

**Sebastien Gouezel: Moment bounds for non-uniformly expanding maps**

*The growth rate of moments of Birkhoff sums is commonly used in physics to estimate specific parameters of dynamical systems. We give a theoretical justification to this process, in intermittent maps or more generally in Young towers: we get precise (matching) upper and lower bounds for this growth rate, in the whole range of parameters of interest, and deduce that the growth rate indeed encodes how non-uniform the expansion is. The methods involve precise dynamical estimates and martingale inequalities.*

**Jacek Graczyk: Mean wiggly compacts**

*We study lower and upper bounds of the Hausdorff dimension for sets which are wiggly at scales of positive density. The continuum is mean wiggly if one can see that it zigzags at positive density of scales for most of its points. The theory of mean wiggly continua leads to new estimates of the Hausdorff dimension for compact sets. We prove also that asymptotically flat sets are of Hausdorff dimension 1 and that asymptotically non-porous continua are of the maximal dimension. Mean wiggly continua are dynamically natural as they occur as Julia sets of quadratic polynomials for parameters from a generic set on the boundary of the Mandelbrot set. The theory can be used to find non-trivial bounds on the Hausdorff dimension of generic Julia sets for small perturbations of the Tchebyshev quadratic polynomial. (Joint work with P. Jones and N. Mihalache)*

**Nicolai Haydn: Limiting distribution for returns to balls in non-uniformly hyperbolic systems**

*We show that for systems that allow a Young tower construction with polynomially decaying correlations the return times to metric balls are in the limit Poisson distributed. We also provide error terms which are powers of logarithm of the radius. In order to get those uniform rates of convergence the balls we have to avoid a set whose size is estimated to be of similar order. This result can be applied to non-uniformly hyperbolic maps and to any invariant measure that satisfies a weak regularity condition. In particular it shows that the return times to balls is Poissonian for SRB measures on attractors thus extending a previous result of Chazottes and Collet to polynomially decaying correlations.This is joint work with K Wasilewska.*

**Sandra Hayes: The Real Dynamics of Bieberbach’s Example**

*Bieberbach constructed in 1933 domains in C2 which were biholomorphic to C2 but omitted an open set. The existence of such domains was unexpected, because the analogous statement for the one-dimensional complex plane is false. The special domains Bieberbach considered are given as basins of attraction of a cubic Hénon map. This classical method of construction is one of the first applications of dynamical systems to complex analysis.*

In this talk the boundaries of the real sections of Bieberbach’s domains will be shown to be smooth. The real Julia sets of Bieberbach’s map will be calculated explicitly and illustrated with computer generated graphics.

In this talk the boundaries of the real sections of Bieberbach’s domains will be shown to be smooth. The real Julia sets of Bieberbach’s map will be calculated explicitly and illustrated with computer generated graphics.

**Mike Hochman: Dimension of self-similar measures via additive combinatorics**

*I will discuss recent progress on understanding the dimension of self-similar sets and measures. The main conjecture in this field is that the only way that the dimension of such a fractal can be "non-full" is if the semigroup of contractions which define it is not free. The result I will discuss is that "non-full" dimension implies "almost non-freeness", in the sense that there are distinct words in the semigroup which are extremely close together (super-exponentially in their lengths). Applications include resolution of some conjectures of Furstenberg on the dimension of sumsets and, together with work of Shmerkin, progress on the absolute continuity of Bernoulli convolutions. The main new ingredient is a statement in additive combinatorics concerning the structure of measures whose entropy does not grow very much under convolution. If time permits I will discuss the analogous results in higher dimensions.*

**Huyi Hu: Some ergodic and rigidity properties of Heisenberg group actions**

*We consider a smooth Heisenberg group action $\mathcal H$ on a manifold $M$, that is, a set of maps of $M$ of the form $f^ng^mh^k$, where $f$, $g$ and $h$ are $C^2$ diffeomorphisms such that $fh=hf$, $gh=hg$ and $fg=gfh$. We give that all Lyapunov exponents of $h$ are zero with respect to any probability measure invariant under $f$, $g$ and $h$. If any map in the group is a codimensional one Anosov diffeomorphism, then $h^k=\text{id}$ for some integer $k$, hence, the action cannot be faithful. We also get some rigidity results similar to those for Abelian group action, that is, if $M$ is a torus, and $H$ contains a hyperbolic element, then there is a homeomorphism $\phi$ such that every element in $\mathcal H$ is topologically conjugate to an affine map of $M$ through $h$.This is a work in progress joint with Enhui Shi and Zhenqi Wang.*

**Francois Ledrappier: Regularity of the stochastic entropy under conformal changes**

*We consider a negatively curved closed Riemannian manifold.The "stochastic entropy" and the "linear drift" are two numbers which describe how the heat diffuses at infinity on the universal cover in large times. We recall some of their properties and discuss their regularity as functions of the metric. This is joint work with Lin Shu.*

**Lingmin Liao: Dirichlet uniformly well-approximated numbers**

*For a fixed irrational $x$, we consider the numbers $y$ satisfying that for all large number $Q\gg1$, there exists an integer $1\leq n\leq Q$, such that the distance from $nx-y$ to the nearest integer is less than $Q^{-s}$, with $s>1$. These numbers are called Dirichlet uniformly well-approximated numbers by the reason of the classical uniform Dirichlet Theorem in Diophantine approximation. For any $s>1$, the Haussdorff dimension of the set of these numbers is obtained and is shown to depend on the Diophantine approximation property of $x$. This is a joint work with Dong Han Kim.*

**Ian Melbourne: The Lorenz attractor is rapid mixing, and consequences**

*We prove that every geometric Lorenz attractor has superpolynomial decay of correlations with respect to the unique SRB measure. Moreover, we prove the central limit theorem and almost sure invariance principle for the time-1 map of geometric Lorenz attractors. In particular, our results apply to the classical Lorenz attractor. (Joint work with Vitor Araujo and Paulo Varandas)*

**Jorge Miguel Milhazes de Freitas: Extreme value laws and point processes of rare events for dynamical systems**

*We will make a brief introduction to the theory of extreme value laws in the context of dynamical systems. The existence of such limiting laws is connected to the study of recurrence and, in particular, to the existence of Hitting Times and Return Times Statistics. We will consider point processes of rare events and show its convergence to the Poisson process or to a compound Poisson process, depending on whether we are in the absence or presence of clustering of extreme events. Moreover, we will show that if the rare events correspond to small targets around non periodic points then no clustering occurs. On the other hand, if the targets are set around repelling periodic points, there will be clustering whose intensity depends on the expansion rate at that point. Assuming that the systems have a strong form of decay of correlations, namely, summable decay of correlations of observables in a suitable Banach space against all L^1 observables, then there exists a dichotomy corresponding to those two limits for the point processes considered.*

**Matthew Nicol: Annealed and quenched limit theorems for random expanding dynamical systems**

*We investigate annealed and quenched limit theorems for random expanding dynamical systems. Making use of functional analytic techniques and more probabilistic arguments with martingales, we prove annealed versions of a central limit theorem, a large deviation principle, a local limit theorem, and an almost sure central limit theorem. We also discuss the quenched central limit theorem, dynamical Borel-Cantelli lemmas, Erd\"{o}s-R\'enyi laws and concentration inequalities. Applications include settings where the chosen maps may include non-uniformly expanding and intermittent type maps. (Joint with Romain Aimino and Sandro Vaienti)*

**Mark Pollicott: Analytic dependence of Hausdorff dimension**

*We show how to demonstrate the analytic dependence of the Hausdorff Dimension of real analytic two dimensional horseshoes via properties of periodic points. This has interesting applications to discrete Schroedinger equations. Time permitting, we will also discuss other problems of derivatives of dimension related to problems for Fuchsian groups.*

**Feliks Przytycki: On the Lyapunov spectrum in 1-dimensional dynamics**

*I plan to talk on the Hausdorff dimension spectrum for Lyapunov exponents, in particular on the function $\alpha\mapsto HD\{x\in K: \lim_{n\to\infty} 1/n \log|(f^n)'(x)|=\alpha\}$, where either $K=J$ is Julia set for a rational map $f$ on the Riemann sphere, or $K$ is the maximal invariant set for a generalized multimodal map on a domain in $R$, provided $f|_K$ is topologically transitive of positive topological entropy.These results have been obtained jointly with Katrin Gelfert and Michal Rams and the real 1-dimensional background comes from a preprint by the speaker and Juan Rivera-Letelier. If time allows I will sketch a proof that for $f$ rational with a critical point $c\in J$ being the only critical point with its forward trajectory accumulating in $J$, the lower Lyapunov exponent at the critical value, namely $\liminf_{n\to\infty} 1/n \log |(f^n)'(f(c))|$, is non-negative (the result joint with Genadi Levin and Weixiao Shen).*

**Federico Rodriguez Hertz: Rigidity of hyperbolic higher rank lattice actions**

*I will discuss some recent results with Aaron Brown and Zhiren Wang on actions by higher rank lattices on nilmanifolds. I will present the result in the simplest case possible, SL(n,Z) acting on Tn, and try to present the ideas of the proof. The result imply existence of invariant measures for SL(n,Z) actions on Tn with standard homotopy data as well as global rigidity of Ansosov actions on infranilmanifolds and existence of semiconjugacies without assumption on existence of invariant measure.*

**Benoit Saussol: Dimension and recurrence for some dynamical systems with polynomial decay of correlation.**

*As the pointwise dimension of a measure, return times into small balls are also related to the Hausdorff dimension.For several class of dynamical systems it is known that the recurrence rate is a.e. equal to the dimension of the measure. This includes the class of systems with a super-polynomial decay of correlations. On the contrary, there are some counter-examples for some parabolic systems with polynomial decay of correlations. The main result of this talk is that if a system can be modeled by a Young tower with a polynomial rate then the recurrence rate is always equal to the dimension. This is a joint work with Francoise Pene (Brest).*

**Dierk Schleicher: Hausdorff Dimension and Paradoxes in Transcendental Dynamics**

*For iterated holomorphic maps in the complex plane, the dynamical plane naturally decomposes into the set I consisting of those points that converge to infinity under iteration and the complement. Both sets are invariant and non-empty. For many transcendental entire functions, these sets are quite interesting from the point of view of Hausdorff dimension. We present a "paradoxical" example that yields in a most natural way a decomposition of the complex plane into two sets R (rays) and E (endpoints), so that each component in R (each ray) connects exactly one point in E (its endpoint) to infinity, and each point in E is connected to infinity by one or more rays. The set R of all rays has Hausdorff dimension 1 --- so in a very strong sense almost every point is in E, and yet each point in E has its own ray(s) that connect the point to infinity! The decomposition arises naturally in the sense that each ray R is (essentially) a path component of the set I. --- This example sharpens an earlier "paradox" by Karpinska where R has dimension 1 and E has dimension 2 (but both have zero Lebesgue measure).*

**Masato Tsujii: Spectrum of geodesic flow on negatively curved manifold**

*We consider the one-parameter families of transfer operators for geodesic flows on negatively curved manifolds. We show that the spectra of the generators have some "band structure" parallel to the imaginary axis. As a special case of "semi-classical" transfer operator, we see that the eigenvalues concentrate around the imaginary axis with some gap on the both sides. Those eigenvalues appear as the zeros of the so-called semi-classical (or Gutzwiller-Voros) zeta functions. These results are obtained as application of some ideas in the semi-classical analysis.*

**Christian Wolf: Localized pressure and equilibrium states**

*In this talk we introduce a natural notion of localized topological pressure and discuss several of its variational properties. We derive the local variational principle for a wide variety of spaces and potentials but also obtain counterexamples. Next, we discuss localized equilibrium states and show that even in the case of systems and potentials with strong thermodynamic properties the classical theory of equilibrium states breaks down. The results presented in this talk are joint work with Tamara Kucherenko.*

**Hongkun Zhang: Optimal bound of mixing rates for nonuniformly hyperbolic systems**

*We obtain optimal bound for rates of convergence to SRB measures for nonuniformly hyperbolic systems with singularities, which include chaotic billiards. Our results are applied to several examples to obtain optimal polynomial mixing rates with order n^{-a}, with a>0. This is a joint work with Sandro Vaienti (CPT) and Nikolai Chernov (UAB).*

**Michel Zinsmeister: Minkowski dimension of quasicircles**

*Let J_t be the Julia set of the polynomial z^2+t and d(t) its Hausdorff dimension. Using thermodynamic formalism Ruelle has proven that d is a real analytic function and he computed d''(0). Recently Mc Mullen has generalized this theorem and found the second derivative in many dynamical situations leading to a perturbation of the circle. This second derivative makes sense even for non dynamical situations and Mc Mullen has asked if the formula remains true in general. The aim of this talk is to show one counter example and a large set of examples where it is true. (Joint work with Le Thanh Hoang Nhat)*

## Posters

Poster session: Aimino, Romanowska, Volk:

Julia Romanowska:

Denis Volk:

*Concentration inequalities for deterministic, random and non-autonomous dynamical systems*Julia Romanowska:

*Measure and Hausdorff dimension of randomized Weierstrass-type functions*Denis Volk:

*Piecewise translations with overlaps*