Schedule

Title: Minimality of strong unstable foliations

Abstract: The unstable foliation of a mixing Anosov diffeomorphism is minimal. In this talk, we will focus on laminations that are tangent to a strong unstable bundle, for uniformly hyperbolic or partially hyperbolic systems: we will discuss the minimality of the laminations, their stability by perturbation. (Joint works with A. Avila, A. Wilkinson.) If time permits I will present some applications.

Title: Block conjugacy of Irreducible Toral Automorphisms.

Abstract: We establish a matrix characterization (called block conjugacy) of the notion of weak equivalence of ideals. This gives the existence of a one-to-one correspondence between equivalence classes of block conjugate toral automorphisms (of a given irreducible characteristic polynomial) and equivalence classes of the weakly equivalent associated ideals. This 1-1 correspondence is analogous to the classical Latimer-MacDuffee-Taussky Theorem and generalizes a result of Laffey. We show that there exist nonconjugate irreducible hyperbolic toral automorphisms A and B for which A+A and B+B are conjugate. This is joint work with Pedro Martins Rodrigues. 

Title: Measures of maximal entropy that are SRB

Abstract:  How often does it occur that the measure of maximal entropy of a system is an SRB measure? We study this question for C1+α partially hyperbolic diffeomorphisms isotopic to Anosov (DA-difeomorphisms) on T3, and establish a rigidity result: the measure of maximal entropy is an SRB measure if and only if the sum of its positive Lyapunov exponents coincides with that of the linear Anosov map A on all periodic orbits of the support of the measure. In that case, the measure is also the unique physical measure.

We also show non-Anosov examples satisfying this condition, both in the conservative and in the non-conservative setting. Finally, we prove that a volume preserving C2 DA-diffeomorphism on T3 is Anosov if all Lyapunov exponents coincide almost everywhere with those  of the linear Anosov in the isotopy class. As a consequence, a smooth DAdiffeomorphism is smoothly conjugated to its linear part if and only if all Lyapunov exponents coincide almost everywhere with those of its linear part.

Title: Getting by without product structure

Abstract: Local product structure is one of the mechanisms driving many fundamental phenomena in hyperbolic dynamical systems, but these phenomena can continue even when local product structure fails; some of Todd Fisher's earliest work was devoted to studying such examples. I will describe some natural examples in which there are obstructions to using arguments based on local product structure, but in which specification-type arguments still lead to results such as uniqueness of the measure of maximal entropy.

Title: Exponential mixing MME for some diffeomorphisms with dominated splitting

Abstract: Todd Fisher studied measures maximizing the entropy (or MMEs) for classical diffeomorphisms satisfying partial hyperbolicity or dominated splitting. I will revisit some of these works using the notion of strong positive recurrence recently developed by S. Crovisier,  O. Sarig, and myself - especially in its characterization as a property of Lyapunov exponents. In the easiest cases, we obtain exponential mixing  from the estimates in Todd's works. Other cases use the invariance principle. Many questions remain open. 

Title: Measures of maximal entropy for discretized Anosov flows

Abstract: In a joint  work with Jérôme Buzzi and Todd Fisher we proved  that time-one maps of transitive Anosov flows of compact manifolds are accumulated by diffeomorphisms robustly satisfying the following dichotomy: either all of the measures of maximal entropy are non-hyperbolic, or there are exactly two ergodic measures of maximal entropy, one with a positive central exponent and the other with a negative central exponent. Our approach was based on an approach by Margulis and our result was proved in a larger class than just close to time one map of transitive Anosov flows. However, in another work joint with Jérôme Buzzi, Sylvain Crovisier and Mauricio Poletti we generalized the previous results and proved a more precise dichotomy for all Discretized Anosov flows: either two ergodic hyperbolic maximal measures or exactly one maximal measure which is non-hyperbolic.

Title: Hornbilliards

Abstract: Recently, Mark Levi and I discovered a mysterious connection between centralizers and the dynamics of cusps in "heavy" billiards. It suggests an answer to whether "horns" in billiards permit infinitely many collisions in finite time which means that these may not have to be excluded from the standard theory of hyperbolic billiards.

Title: Entropy, partial hyperbolicity and the work of Todd Fisher

Title: Absolute continuity of stationary measures

Abstract: In this talk, we will explore smooth random dynamical systems on surfaces. Aaron Brown and Federico Rodriguez Hertz proved that hyperbolic  stationary measures have the SRB property, except when certain obstructions occur. The SRB property here (essentially) means that the measure is absolutely continuous along certain ``nice'' curves (unstable manifolds). In this talk, we want to identify conditions under which the stationary measure is absolutely continuous with respect to the Lebesgue measure of the ambient space. Our approach is inspired by  Tsujii's "transversality" method used to prove Palis Conjecture for partially hyperbolic surface endomorphisms.  This is a joint work with Aaron Brown, Homin Lee and Yuping Ruan. 

Title: Equilibrium States in Partially Hyperbolic Sets: some examples and results. 

Abstract: This lecture presents  two  contributions to the theory of equilibrium states in partially hyperbolic systems. The first paper, "Equilibrium States for Certain Partially Hyperbolic Attractors",  in collaboration with Todd Fisher, we examine the unique equilibrium states for a class of partially hyperbolic attractors, utilizing techniques developed by Climenhaga and Thompson. The second result, "Uniqueness of Equilibrium Measure for a Family of Partially Hyperbolic Horseshoes"  jointly with Marlon Oliveira and Eduardo Santana, focuses on the uniqueness of equilibrium states for partially hyperbolic horseshoes using symbolic dynamics. Time permitting, we pose some questions that we do not know how to answer.