I will give a talk: **SRB measures without symbolic dynamics or dominated splittings** on **Thursday, June 9th**.

This is joint work with Yakov Pesin and Dmitry Dolgopyat, in which we examine sufficient conditions for a non-uniformly hyperbolic system to have an SRB measure. We introduce a notion of *usable hyperbolicity* and use a new version of the Hadamard-Perron theorem to show that if there is a positive volume set of points with positive average asymptotic rate of usable hyperbolicity, then the system has an SRB measure.

We can apply this approach to certain examples obtained by perturbing an Axiom A system near a fixed point (as in the Katok example, and also by adding a shear). We would like to understand if this approach can be applied to other interesting classes of examples, such as one-parameter families of perturbations of non-uniformly hyperbolic systems.

The preprint is available, and the slides are here.

My other main project at the moment is joint work with Daniel Thompson, in which we study systems with a weaker version of the specification property and give sufficient conditions for uniqueness of an equilibrium state.

- In this preprint (to appear in
**Israel J. Math.**), we study uniqueness of the measure of maximal entropy and show in particular that every subshift factor of a beta-shift has a unique measure of maximal entropy, settling an open problem of Klaus Thomsen that appeared in Mike Boyle's**Open problems in symbolic dynamics**. - In the sequel to that paper (preprint coming soon, slides available here), we extend our method to equilibrium states, and show in particular that every Holder continuous potential function on a beta-shift has a unique equilibrium state.

My homepage is here.