# Titles and Abstracts

Jon Aaronson

Title: Rational weak mixing in infinite measure spaces

Abstract: Motivated by an example of E. Hopf (1937), K. Krickeberg (1967) introduced a topological ratio mixing property which is enjoyed by Markov shifts with the strong ratio limit property.

The theory of weakly wandering sets as developed by Hajian and Kakutani (1964) shows that there is no measure theoretic version of this ratio mixing property.

Nevertheless, Hopf's example is "rationally weak mixing" in a sense which implies that for every pair of measurable sets in a hereditary ring, ratio mixing is satisfied along a subsequence of density one. If there's time, we'll also discuss some questions arising for Markov shifts and other infinite measure preserving transformations.

Keith Burns

Title: On the dimension of the level sets for the Lyapunov exponent of a geodesic flow

Abstract: The talk describes joint work with Katrin Gelfert. We study the Hausdorff dimension of the level sets for the Lyapunov exponent of the geodesic flow of a compact surface with nonpositive curvature.

Gonzalo Contreras

Title: Generically the 2-sphere has an elliptic closed geodesic.

Abstract: We prove that there is an open and dense set in the C2 topology of riemannian metrics on the 2-sphere whose geodesic flow has an elliptic closed geodesic.

This result recovers, in a generic sense, a theorem by H. Poincaré 1904 and proves a conjecture by Michel Herman 2000. The proof combines techniques in symplectic dynamical systems (stable hyperbolicity) and contact geometry.

Sebastian Ferenczi

Title: Ergodic generalizations of the Lagrange spectrum

Abstract: We compute two invariants of topological conjugacy, the upper and lower limits of the inverse of Boshernitzan's *ne _{n}*, where

*e*is the smallest measure of a cylinder of length

_{n}*n*, for three families of symbolic systems, the natural codings of rotations and three-interval exchanges and the Arnoux-Rauzy systems. The sets of values of these invariants for a given family of systems generalize the Lagrange spectrum, which is what we get for the family of rotations with the upper limit of .

Ilya Goldsheid

Title: Products of random transformations and random walks in random environments

Abstract: Random transformations arise naturally in the context of the study of random environments. I shall explain some recent results about the latter in terms of the properties of the former.

Genadi Levin

Titile: On the problem of "wild attractor" in one-dimensional dynamics.

Abstract: We discuss the above problem in the context of unimodal maps (real and complex) and critical circle covering maps. Based on joint works with Greg Swiatek.

Michal Misiurewicz

Title: Concavity implies attraction

Abstract:

We consider a skew product with the interval [0,a] as a fiber space and maps in fibers that are concave and fix 0. If the map in the base is an irrational rotation of a circle, then it has been known that under some additional conditions there exists a Strange Nonchaotic Attractor (SNA) for the system. The proofs involved Lyapunov exponents and Ergodic Theorem. We show that the existence of an attractor basically follows only from the uniform concavity of the maps in the fibers. In particular, it does not depend on the map in the base, so it occurs also in a nonautonomous case. Next, we discuss the possible generalizations of the notion of a SNA and show the problems that can occur in the case when the map in the base is noninvertible. This is a joint work with Lluis Alseda.

Feliks Przytycki:

Title: Some of Anthony Manning's mathematical achievements

Abstract: We will discuss the entropy conjecture, dimension of maximal measure and dimension of horseshoes - topics on which Anthony Maning has made important contributions.

Mary Rees

Title: Followers are back

Abstract: In studies of the prevalence of non-uniform hyperbolicity, a very useful idea is to introduce symbolic dynamics and then use the low density of words with proportionately long followers: that is, in most words over a finite alphabet, the initial segment of some length *n* is not repeated in the initial segment of length (say) . I shall describe how such an idea arises in a rather different setting in complex dynamics, when the problem is to determine how many different representations of a standard type, up to topological conjugacy, a hyperbolic rational map can have.

Omri Sarig

Title:

Abstract: Let f be *C*^{1+α} diffeomorphism on a compact smooth surface, and suppose the topological entropy h(f) is positive.

For every 0<ε<h(f) we construct an invariant Borel set E s.t.

(1) f|E has a countable Markov partition

(2) E is large in the sense that m(E)=1 for every ergodic invariant probability measure m with entropy larger than ε.

In the first hour I will explain what this means, and what this implies.

In the second hour I will give a sketch of the proof.

Weixiao Shen

Title: Stochastic stability of non-uniformly expanding interval maps

Abstract: We study random perturbations of an interval map f under the following non-uniformly expanding condition: for each critical value v, ∑_{n=0}^{∞} |*Df ^{n}*(

*v*)|

^{-1}<∞. We shall prove strong stochastic stability for such a map under a mild condition on transition probabilities.

Károly Simon TU Budapest

Title: Algebraic difference of random Cantor sets

Abstract: Palis and Takens studied the unfolding of homoclinic tangency in some one parameter families of diffeomorphisms on the plane. The set of those parameters for which homoclinic tangency occurs can be characterized as the algebraic difference of two dynamically defined Cantor sets on the line. The algebraic difference of the sets *F*_{1},*F*_{2} is defined by

*F*_{2}-*F*_{1}={*y*-*x*: *x*∈*F*_{1}, *y*∈*F*_{2}}

Palis conjectured that if

dim_{H }*F*_{1}+ dim_{H} *F*_{2}>1

then *generically* it should be true that

*F*_{2}-*F*_{1} contains an interval.

In this talk we consider the same problem for some families of random Cantor sets. The talk is based on a recent joint work with M. Dekking and B. Székely.

Yakov Sinai

1. Bifurcations in solutions of 2-dim Navier-Stokes system

2. Mobius function and statistical physics

Philippe Thieullen

Title: Zero-temperature Gibbs measures for some subshifts of finite type

Abstract: A zero-temperature Gibbs measure is any limit measure of Gibbs measures obtained as the temperature goes to zero. For locally finite energy, the limit is known to exist. For general Holder energy, there may exist several accumulation points. When the ground-state configuration set has a unique irreducible component of maximal topological entropy, we show the convergence of the Gibbs potential as the temperature goes to zero.

Cecilia Tokman

Title: Semi-invertible multiplicative ergodic theorems

Abstract: Semi-invertible multiplicative ergodic theorems establish the existence of an Oseledets splitting for cocycles of non-invertible linear operators (such as transfer operators) over an invertible base. Using a constructive approach, we establish a semi-invertible multiplicative ergodic theorem, and give an application to Perron-Frobenius operators of random composition of maps. Joint work with Anthony Quas

Andrew Torok

Title: Convergence of moments for Axiom A and nonuniformly hyperbolic flows

Abstract: We prove convergence of moments of all orders for Axiom A diffeomorphisms and flows. The same results hold for nonuniformly hyperbolic

diffeomorphisms and flows modelled by Young towers with superpolynomial tails. For polynomial tails, we prove convergence of moments up to a certain order, and give examples where moments diverge when this order is exceeded.

Nonuniformly hyperbolic systems covered by our result include Henon-like attractors, Lorenz attractors, semidispersing billiards, finite horizon planar periodic Lorentz gases, and Pomeau-Manneville intermittency maps.

This is a joint work with Ian Melbourne, University of Surrey.

Serge Troubetzkoy

Title: Recurrence of the periodic Ehrenfest wind tree model (periodic planar billiards)

Abstract: Consider the plane, and remove an identical rectangle centered at each point of *Z*^{2}. Play billiards in the infinite table which is the complement of the rectangles. We show almost sure recurrence of the billiard. This is joint work with Pascal Hubert and Samuel Lelievre.

Sandro Vaienti

Title: Polynomial Decay for Non Markov Maps

Abstract: TBA

Edson Vargas

Title: Dirac physical measures and transitive flows

Abstract: We discuss some examples of smooth transitive flows with physical measures supported at fixed points. Using the Anosov-Katok method, we construct transitive flows on surfaces with the only ergodic invariant probabilities being Dirac measures at hyperbolic fixed points. When there is only one such point, the corresponding Dirac measure is necessarily the only physical measure with full basin of attraction. Using an example due to Hu and Young, we also construct a transitive flow on a three-dimensional compact manifold without boundary, with the only physical measure the average of two Dirac measures at two hyperbolic fixed points. This is part of a joint work with Radu Saghin and Wenxiang Sun.

Anatoly Vershik vershik@googlemail.com

Title: Dynamics of the metric in ergodic theory and scaling entropy

Abstract: We consider the evolution of the so-called admissible metrics on the measure space and its asymptotic behavior under the action of the group or under the rules related to dynamics. One of the powerful invariants of such dynamic of the metric is rate of epsilon-entropy of the metric. This invariant allows to distinguish the different kind of the filtration (=decreasing sequence of the sigma-fields) and metric type of automorphisms.

**SHORT TALKS: **

Carlo Carminati (with Giulio Tiozzo): The family of generalized bounded type numbers and its bifurcation locus

Jianyu Chen: Coexistence of zero and nonzero Lyapunov exponents

Abstract: The persistence of invariant tori in the category of volume-preserving diffeomorphisms was established by C.-Q. Cheng, Y.-SSun, M. Hermann, Z.Xia, J-C. Yoccoz. Outside those "elliptic islands" , it is possible that there are "chaotic sea" with (nonuniformly) completely hyperbolic behavior. I shall introduce some results in this direction. H. Hu, Ya. Pesin and A. Talitskaya constructed a volume-preserving diffeomorphism of a 5-dim manifold, which is ergodic and has nonzero Lyapunov exponents on an open dense subset of not full measure， and has zero exponents on the complement consisting of codim-2 invariant submanifolds. With careful modifications, we can construct a similar example but with countably many ergodic components

Carl Dettmann: The Lorentz gas and the Riemann Hypothesis

Abstract: The Lorentz gas is a model of deterministic diffusion consisting of many fixed scatterers with which a point particle makes elastic ("billiard") collisions. In a two dimensional square lattice of circular scatterers it is known that displacements are normally distributed, but with a nonstandard, superdiffusive scaling of the form sqrt(t ln t). I will describe attempts to extend this result to higher dimensions. In the case of small scatterers, the superdiffusion coefficient is related to the Riemann hypothesis. There is also theoretical and numerical evidence for a qualitative change of behaviour in six dimensions and beyond.

Dong Han Kim: Diophantine condition of the interval exchange map

Abstract: The interval exchange map is a generalization of the irrational rotation. The irrational rotation can be classified by the Diophantine type of its rotation angle. In this talk, I will discuss how to generalize the Diophantine condition for the interval exchange map. It can be obtained by the size of the continued fraction matrix,the recurrence time, and the distance between discontinuities for the iterated map. We also consider the relation between them. This is a joint work with Stefano Marmi.

David Simmons: Random Iteration of Rational Functions

Abstract: It is a theorem of Denker and Urbanski that if *T *is a rational map of degree at least two and if is Holder continuous and satisfies the "thermodynamic expanding'' condition , then there exists exactly one equilibrium state μ for *T *and ϕ, and furthermore is metrically exact. We extend these results to a random setting by defining the notion of a random holomorphic action on . Pressure and entropy are as defined by Ledrappier and Walters in their paper which proves a variational principle for random dynamical systems.

Maciej P. Wojtkowski: On the real baker map

We provide a mathematical model for the mixing protocol used traditionally by bakers in the process of kneading dough. The model is 3-dimensional in contrast to the popular 2-dimensional version, but they share many properties. In particular the map is isomorphic to the Bernoulli shift on two symbols. We also describe the speed of mixing in geometric terms.

Christian Wolf: The geometry of rotation sets

Abstract: Let $T:X\to X$ be a continuous transformation on a compact metric space, and let $\phi_1,...,\phi_n$ be continuous observables. The rotation set $R_T(\phi_1,...,\phi_n)$ is the set of all $\mu$-integral vectors where $\mu$ runs over all invariant probability measures. It is easy to see that the rotation set is a compact, convex subset of $R_n$. In particular, it has a Lipschitz boundary. In this talk we discuss the opposite question, namely is every set with these properties attained as a rotation set of a particular set of potentials within a particular class of dynamical systems. We give a positive answer in the case of shift maps and also provide criteria implying that the boundary of the rotation set is piecewise smooth. The results in this talk are joint work with Tamara Kucherenko.