Statistical Properties of Generalized Horseshoe Maps

TL;DR

This paper uses thermodynamic formalism to analyze a generalized horseshoe map, establishing spectral properties of the transfer operator and conditions for the existence of a unique physical measure with specific regularity.

Contribution

It introduces a tailored anisotropic Banach space framework that guarantees a spectral gap and characterizes the physical measure's regularity under expanding conditions.

Findings

Spectral gap for the transfer operator established

Existence of a unique physical measure proven

Physical measure is absolutely continuous with Sobolev regularity

Abstract

We apply thermodynamic formalism to a generalized horseshoe map. We prove that a tailored anisotropic Banach space with weighted norms yields a spectral gap for the transfer operator, implying the existence of a unique physical measure. Under the virtually expanding condition, this measure is absolutely continuous with respect to Lebesgue measure, with density in the Sobolev space $H_\mu$, for some $\mu<1/2$.