Paulo Varandas (Universidade de Aveiro) and Nicola Bertozzi (Università di Pisa)

Abstract: In this talk, we present a broad picture of the statistical behavior of a class of chaotic flows with codimension one Axiom A attractors. Our goal is to understand when these systems exhibit strong mixing properties and robust statistical laws, building on the groundbreaking work of Dolgopyat, as well as criteria for exponential decay of correlations developed by Avila, Gouëzel, and Yoccoz, and more recently by Araújo and Melbourne. Assuming a mild regularity condition on the stable foliation, I will discuss a dichotomy: either the system displays exponential decay of correlations for a wide class of natural invariant Gibbs measures, or its stable and unstable directions satisfy a rigid geometric compatibility. This talk is based on joint work with D. Daltro.

Abstract: It is a well-known fact in dynamical systems that the geodesic flow on the modular surface exhibits exponential decay of correlations (Ratner 1986), the proof involving algebraic and representation theory-related tools. In 1998, Chernov explored the possibility of showing the same result through a more classical and general dynamical argument, involving Markov partitions. Later, Dolgopyat proposed a method for proving exponential decay of correlations for geodesic flows on negatively curved compact surfaces. The talk shows how to adapt this method to the case of the flow on the modular surface, which is not compact. In particular, this flow is conjugated to a specific suspension flow (Bonanno, Del Vigna, Isola, 2024): we study a generalization of this system. Then, we address the possibility of removing the geometric framework by considering suspension flows over Markov subshifts with countably infinite alphabets. Under suitable conditions, we recover exponential decay of correlations for these symbolic flows acting on non-compact phase spaces.