SRB measures for uniformly expanding on average partially hyperbolic systems
Davi Obata (Brigham Young University)

A central conjecture of Jacob Palis asserts that a typical dynamical system admits finitely many SRB measures describing the asymptotic statistical behavior of almost every point. For partially hyperbolic diffeomorphisms, fundamental works of Alves, Bonatti, and Viana established conditions ensuring existence and finiteness of SRB measures in the mostly expanding or mostly contracting settings. However, much less is known when the central direction exhibits mixed behavior, with regions of nonuni- form expansion and contraction.

In this talk, I will introduce a weak form of hyperbolicity, called uniform expansion on average, which nevertheless implies several strong ergodic properties. This condition arises naturally in many examples with mixed central behavior and leads to the existence and finiteness of SRB measures, exponential mixing (up to a period) of SRB measures, and continuity of Lyapunov exponents. In this talk, I will focus on the exponential mixing property for such measures. This is joint work with Pablo D. Carrasco, Radu Saghin, and Jiagang Yang.