On statistical properties for equilibrium states of partially hyperbolic horseshoes

TL;DR

This paper investigates the statistical properties of equilibrium states in partially hyperbolic horseshoes, establishing spectral gaps, decay of correlations, and a central limit theorem for these complex dynamical systems.

Contribution

It introduces a projection map and proves a spectral gap for its transfer operator, leading to new statistical results for partially hyperbolic horseshoes.

Findings

Spectral gap for the transfer operator on Hölder observables

Exponential decay of correlations in the system

Central limit theorem for equilibrium states

Abstract

We derive some statistical properties for equilibrium states of partially hyperbolic horseshoes. We define a projection map associated to the horseshoe and prove a spectral gap for its transfer operator acting on the space of H\"older continuous observables. From this we deduce an exponential decay of correlations and a central limit theorem. We finally extend these results to the horseshoe.