In this talk, based on joint work with Lima and Melbourne, we discuss how the ideas of Chernov for the construction of Young towers for piecewise smooth hyperbolic systems can be explored to yield similar structures for the geodesic flows on certain non-positively curved surfaces. In particular, by combining this fact with the results by Balint, Butterley, Melbourne, Terhesiu, Torok (among others), we derive several statistical consequences for this class of geodesic flows (including an effective rate of mixing and a central limit theorem). 

For a surface diffeomorphism f I will describe some natural geometric conditions that guarantee existence of a Young tower for f. The base of the tower is a ``rectangle'' which is ``maximal'' with respect to T-returns for some T and the inducing time of the tower is the T-return to its base. As a corollary one obtains existence of an SRB measure for f. This is a joint work with Vaughn Climenhaga and Stefano Luzzatto.

The top Lyapunov exponent and the entropy are well-known to depend discontinuously on the invariant measures. We show that the two defects in lower semicontinuity are linked for $C^\infty$ smooth surface diffeomorphisms. In particular, for a given sequence of ergodic measures, continuity of the entropy implies that of the exponent. The proof relies on Yomdin theory and a key reparametrization lemma for curves near homoclinic  tangencies. This has a number of consequences.

This is joint work with Sylvain CROVISIER and Omri SARIG.

Establishing sharp bounds on the mixing rates of non-uniformly hyperbolic maps has been an active area of research for some time and Markov partitions in one way or another play an important role in this area. In this talk I will focus on a class of 2D, (non-uniformly) hyperbolic, symplectic (area preserving) maps that exhibit intermittent behaviour near a fixed point. The systems under considerations are examples of what are called "almost Anosov" maps, but in contrast to examples considered previously, in our examples, the derivative of the maps at the neutral fixed point is not the identity matrix and the stable and unstable distributions are tangent. This is the generic behaviour near a neutral fixed point for a 2D symplectic map and such a behaviour complicates considerably the geometrical analysis involved in estimating the mixing rate. I will briefly explain the class of examples and describe the ingredients that lead to proving sharp mixing rates for such systems.
(Joint work with Carlangelo Liverani)

For the hyperbolic systems with singularities, Markov partitions are rather delicate to construct because of the fragmentation of the phase space by singularities. In this talk, we investigate a broad class of uniformly hyperbolic systems with singularities, which includes the Sinai dispersing billiards and their small perturbations due to external forces and nonelastic reflections with kicks and slips. Under the standard hypotheses (H1)-(H4), we establish countable Markov partitions by constructing ``perfect'' hyperbolic product sets with exponential tail. Such structure immediately implies the exponential decay of correlations, the central limit theorem and other advanced stochastic properties. We also obtain Markov partitions for certain nonuniformly hyperbolic systems, which includes the semi-dispersing billiards and the Bunimovich billiards. This is a joint work with Fang Wang and Hong-Kun Zhang.

The goal is to introduce and discuss the strong positive recurrence property for diffeomorphisms of a manifold.
I will show that every C-infinity surface diffeomorphism with positive topological entropy satisfies this property. The proof uses results presented by Jérôme Buzzi in his talk on the continuity of the entropy and exponents. Some consequences of this property will be discussed by O. Sarig later.

This talk is a continuation of the lectures by  Buzzi and Crovisier on  surface diffeomorphisms which are differentiable infinitely many times. 

(1) Jerome proved that such diffeos exhibit "entropy continuity of Lyapunov exponents"

(2) Sylvain showed that entropy continuity of Lyapunov exponents implies a property we call "strong positive recurrence"

(3) I will show that "strong positive recurrence" implies exponential decay of correlations and CLT for mixing  measures of maximal entropy.

The method is to build a countable Markov partition whose transition matrix acts with spectral gap on  a nice Banach space. (Joint work with Buzzi & Crovisier)

In this talk I will focus on systems which display certain hyperbolic features.
Inspired by the work of Lai-Sang Young '98 on constructing SRB measures,
Pesin, Senti, and Zhang '16 introduced maps with inducing schemes of hyperbolic type and effected thermodynamical formalism for such maps, i.e., constructed equilibrium measures for a broad class of potentials.
Particular examples of maps admitting inducing schemes of hyperbolic type include: the H\'enon map at first bifurcation,  the Katok map, and a slow-down of a classical Smale-Williams solenoid. The aim of this talk is to present a comprehensive view on the subject and to describe some new results regarding ergodic properties of such systems, such as decay of correlations, the Central Limit Theorem, the Bernoulli property, etc. This is a joint work with Farruh Shahidi.

We introduce a class of orbits which do not require hyperbolicity. That is, sensitivity to initial conditions may be strictly sub-exponential. We discuss symbolic dynamics of such orbits (namely countable Markov Partitions with a finite-to-one almost everywhere coding, and with summable variations for the C^1 distance of stable and unstable manifolds of chains). We discuss a class of examples where every invariant measure can be coded this way, including the Riemannian volume Vol, while Vol-a.e point has a 0 Lyapunov exponent.