September 24: Semyon Dyatlov (Berkeley)
Title: Bounds on the number of Ruelle resonances
Abstract: Ruelle resonances are complex numbers appearing in expansions of correlations. For contact Anosov flows, we prove a sharp upper bound on the number of Ruelle resonances in strips near the real axis. Our approach follows the work of Faure-Sjostrand, who explained how the inhomogeneous Sobolev spaces used by Baladi, Tsujii, Blank, Gouezel, Keller, and Liverani have a clear microlocal interpretation. The refined bounds are obtained using recent work of Sjostrand-Zworski and Datchev and the speaker on resonance counting. Joint work with K. Datchev and M. Zworski.
October 1: Sebastian Hensel (Chicago)
Title: Realisation and Dismantlability
Abstract: In this talk, I will introduce the notion of a dismantlable graph. These graphs have nice combinatorial properties, in particular their automorphism groups always have fixed cliques. As an application of this concept, I will present a new and elementary proof of the Nielsen realisation theorem for Out(F_n). This is joint work with Damian Osajda and Piotr Przytycki.
October 2: Francesco Cellarosi (UIUC)
Title: Ergodic Properties of Square-Free Numbers
Abstract: An integer is square-free if it is not divisible by the square of any prime. We study binary and multiple correlations for the set of square-free numbers and we construct a dynamical systems naturally associated to them. We prove that this dynamical system has pure point spectrum and it is therefore isomorphic to an ergodic translation on a compact abelian group. In particular, the system is not weakly mixing, and it has zero entropy. The latter results were announced recently by Peter Sarnak and our approach gives an alternative proof. Joint work with Yakov Sinai.
Octobet 15: Dan Thompson (Ohio State)
Title: Intrinsic ergodicity, orbit gluing and S-gap shifts
Abstract: This talk is based on a series of papers by Vaughn Climenhaga (Houston) and myself where we develop a new approach to prove uniqueness of equilibrium measures for dynamical systems with various non-uniform structures. One class of model examples that motivated our results is the family of S-gap shifts. The S-gap shifts are a natural family of symbolic spaces which are easy to define but can be challenging to study! For a fixed subset S of the natural numbers, the corresponding S-gap shift is the collection of binary sequences which satisfy the condition that the length of every run of consecutive 0's is a member of S. Examples are the even shift (we let S be the even numbers) and the prime gap shift (we let S be the prime numbers). The key difficulty in the study of S-gap shifts is a spectacular failure of the Markov property, and we develop techniques to deal with this. These techniques can be adapted to apply to many other interesting dynamical systems beyond the well understood uniformly hyperbolic case (e.g. beta-shifts, interval maps with parabolic fixed points, non-uniformly expanding maps in higher dimensions, some partially hyperbolic examples). I will explain our approach and motivations focusing on the example of S-gap shifts, building up to our result that 'Every subshift factor of an S-gap shift (or a beta-shift) is intrinsically ergodic'. If time permits, I will also describe a brand new result by Climenhaga, myself and Kenichiro Yamamoto (Tokyo Denki University) which establishes a large deviations principle for S-gap shifts.
November 12: Giovanni Forni (Maryland)
Title: On the ergodicity of geodesic flows on compact flat surfaces
Abstract: In this talk we will discuss applications of a complex-analytic criterion for the ergodicity of the geodesic flow on a compact flat surface with conical singularities. This criterion is based on the same ideas as in our proof of the spectral gap of the Kontsevich-Zorich cocycle or the ergodicity of the Teichmueller flow. We will outline an application of this criterion to the case of compact translation surfaces and to the case of compact surfaces with irrational cone points. In the first case, we will explain the proof, obtained by R. Trevino in his PhD thesis, of a Kinchin type ergodicity criterion which improves on Cheung and Eskin quantitative version of Masur's criterion. In the second case, we will outline a (still tentative) proof of ergodicity of the geodesic flow under a simultaneous Diophantine condition on the cone angles.
November 19: John Franks (Northwestern)
Title: Symplectic Surface Diffeomorphisms
Abstract: Suppose M is a compact oriented surface of genus 0. We establish a structure theorem for area preserving diffeomorphisms of $M$ with zero entropy and at least three periodic points. As an application we show that rotation number is defined and continuous at every point of a zero entropy area preserving diffeomorphism of the annulus. Applications of this theorem give insight into the algebraic structure of the group Symp of analytic symplectic diffeomorphisms of M. We show that if G is a subgroup of Symp and G has an infinite normal solvable subgroup, then G is virtually abelian. In particular the centralizer Cent(f) of an infinite order element of Symp is virtually abelian. Another immediate corollary is that if $G$ is a solvable subgroup of Symp then G is virtually abelian. We consider the question of whether Symp satisfies the Tits alternative.
November 26: David Aulicino (Chicago)
Title: Teichmueller Discs with Completely Degenerate Kontsevich-Zorich Spectrum
Abstract: I will introduce the Lyapunov exponents of the Teichmueller geodesic flow on the moduli space of Abelian differentials and give some background on what is known about them. Then I will consider the problem of classifying all SL(2,R) orbits with the property that they have the maximal number of zero Lyapunov exponents with respect to the Kontsevich-Zorich cocycle. It is known that there are two surfaces, one in genus three and one in genus four, whose SL(2,R) orbits have this property. Recent progress toward showing that these are the only two such orbits will be presented.