Title: Thermodynamic formalism of fiber-bunched GL(d,R)-cocycles

Speaker: Kiho Park – University of Chicago

Abstract: We study subadditive thermodynamic formalism of H\”older and fiber-bunched GL(d,R)-cocycles over subshift of finite types. Here, fiber-bunched cocycles refer to cocycles that are nearly conformal. Unlike additive thermodynamic formalism where any H\”older continuous potential has a unique equilibrium state, there are examples of H\”older continuous matrix cocycles with multiple equilibrium states. Restricted to fiber-bunched cocycles, we show that there exists an open and dense subset of cocycles with unique equilibrium states; such open and dense subset consists of typical cocycles first introduced by Bonatti and Viana. The unique equilibrium states of typical cocycles follow from a property known as quasi-multiplicativity, and they have the subadditive Gibbs property.When d=2, we have complete description of cocycles with unique equilibrium states. In particular, irreducible cocycles necessarily have unique equilibrium states, and we provide characterization for reducible cocycles with more than one equilibrium states.

Title: New Progress on the Katok-Spatzier conjecture

Speaker: Kurt Vinhage – Pennsylvania State University

Abstract: We will discuss recent progress on the Katok-Spatzier conjecture, which aims to classify Anosov actions of higher-rank abelian groups under the assumption that there are no nontrivial smooth rank one factors. We develop new techniques to build homogeneous structures from dynamical ones. The remarkable features of the techniques are their low regularity requirements and their use of metric geometry over differential geometry to build group actions. We apply these techniques to obtain a classification result in the totally Cartan setting, where bundles associated to the hyperbolic structure are one-dimensional. Joint with Ralf Spatzier.

Title: Measure Rigidity for Anosov Flows via the Factorization Method

Speaker: Asaf Katz – University of Chicago

Abstract: Using the factorization method, a method pioneered by Eskin and Mirzkhani in their groundbreaking work about measure classification for P-invariant measures over the moduli space of translation surfaces, we show that generalized u-Gibbs states over quantitatively non-integrable Anosov systems are absolutely continuous with respect to the whole unstable manifold.

Title: Asymptotic Pairs Versus Positivity of Entropy

Speaker: Tomasz Downarowicz – Wroclaw University of Technology

Abstract: Consider a dynamical system (X,T) consisting of a compact metric space X and iterates of a self-homeomorphism T of this space. Topological entropy of the system depends on the speed of growth of complexity of orbits. Zero entropy means that this growth is subexponential and positive entropy corresponds to exponential growth. All this seems quite sophisticated and subtle. On the other hand, there is a very simple-minded notion of an asymptotic pair: two points x, y in X are asymptotic pair if their orbits come closer and closer together as time advances. It might seem surprising but this simple concept SUFFICES to distinguish between positive and zero entropy systems. During my talk I will try to familiarize the audience with the main ideas behind this result (which is due to Blanchard-Host-Ruette in one direction and D. and Lacroix in the other). Moreover, in recent time D., Oprocha and Zhang have obtained a similar criterion for positive entropy in actions of any countable abelian group. If time permits. I will briefly say why this case is very different from the Z-case.

Title: Thurston’s Master Teapot

Speaker: Kathryn Lindsey – Boston College

Abstract: When a multimodal self-map of an interval is postcritically finite (PCF), its growth rate (the exponential of its topological entropy) is a special type of algebraic number called a weak Perron number. W. Thurston plotted the set of all Galois conjugates of growth rates of PCF unimodal maps; this visually stunning image revealed that this set has a rich and mysterious geometric structure. Thurston’s Master Teapot is a closely related 3D set. This talk will present some of the basic topological and geometrical properties of these sets. Based on joint work with C. Wu. H. Bray, D. Davis.

Title: Pressure in Symbolic Dynamics

Speaker: Anibal Velozo – Yale

Abstract: Symbolic dynamics is a particularly useful tool to understand smooth systems with certain hyperbolicity; this is typically done via a “coding”, i.e. a Markov partition/section or inducing schemes. Moreover, the thermodynamic formalism of finite state and countable Markov shifts is well understood, and many results can be pushed to the smooth setting via such codings. The plan of the talk is to overview some known results about the thermodynamic formalism of finite state and countable Markov shifts, and to discuss some recent works about the limiting behavior of the pressure of invariant measures in the non-compact setting. If time permits we will also discuss analogous results for flows.

Title: Decay of correlations for isometric extensions of Anosov flows

Speaker: Salman Siddiqi, Michigan

Abstract: I will briefly provide some historical context discussing known results on exponential correlation decay (or exponential mixing) for Anosov flows. I’ll summarize some classical techniques, and sketch a proof that locally accessible isometric extensions of Anosov flows are exponentially mixing under certain conditions – this includes, for example, some classes of frame flows and flows on principal bundles.

Title: Open sets of exponentially mixing Anosov flows

Speaker: Khadim War, IMPA and University of Chicago

Abstract: We prove that an Anosov flow with C^1 stable bundle mixes exponentially whenever the stable and unstable bundles are not jointly integrable. This allows us to show that if a flow is sufficiently close to a volume-preserving Anosov flow and dim(E^s) = 1, dim(E^u) ≥ 2 then the flow mixes exponentially whenever the stable and unstable bundles are not jointly integrable. This implies the existence of non-empty open sets of exponentially mixing Anosov flows. This is based on a joint work with Oliver Butterley.

Title: Global rigidity of the periodic Lyapunov spectrum for geodesic flows of negatively curved locally symmetric spaces

Speaker: Clark Butler, Princeton

Abstract: We show that if a smooth Anosov flow f^{t} is orbit equivalent to the geodesic flow g^{t} of a negatively curved locally symmetric space X of dimension at least three and the Lyapunov spectra of the flow f^{t} at all periodic points are multiples of the corresponding Lyapunov spectra of g^{t} then f^{t} is smoothly orbit equivalent to g^{t}. If f^{t} is itself the geodesic flow of a negatively curved space Y then we further conclude that Y is homothetic to X. We deduce the Mostow rigidity theorem as a corollary.

Here is a list of visiting speakers currently scheduled for Fall semester 2019. All talks are on Thursdays in MW154 at 3.00pm-4.00pm (unless otherwise indicated). More talks will be announced soon.

Sept 5: Clark Butler (Princeton)

Sept 12: Khadim War (IMPA / Chicago)

Oct 3: Salman Saddiqi (Michigan)

Oct 17: Anibal Velozo (Yale)

Oct 24: Kathryn Lindsey (Boston College)

Oct 31: Tomasz Downarowicz (Wroclaw University, Poland)

Nov 7: Asaf Katz (Chicago)

Nov 14: Kurt Vinhage (Penn State)

Nov 21: Kiho Park (Chicago)