Spectral Gaps and Statistical Properties for Low-Regularity Dynamical 

Daniel Smania - ICMC-USP

Abstract:

We present recent results on the spectral analysis of transfer operators acting on Banach spaces of either functions (for piecewise expanding maps) or distributions (for the full shift), constructed using atomic decompositions. In particular, we study transfer operators on Besov spaces associated with piecewise expanding maps under very mild regularity assumptions on the dynamics, the potentials, and the phase space. This leads to quasi-compactness, exponential decay of correlations, and limit theorems for a wide class of observables, including certain unbounded ones (joint work with Alexander Arbieto).

In the symbolic setting, we consider the bilateral shift with Hölder weights and construct Banach spaces of distributions on which the transfer operator is quasi-compact with a spectral gap. The unique Gibbs measure spans the leading eigenspace, yielding exponential mixing for a broad class of reference measures. This is part of the Ph.D. work of Mateus Marra (ICMC-USP, Brazil), under my supervision.