Thermodynamics of the Katok Map

TL;DR

This paper develops thermodynamical formalism for the Katok map, a non-uniformly hyperbolic dynamical system, proving existence and uniqueness of equilibrium measures and establishing statistical properties like decay of correlations and CLT.

Contribution

It introduces a thermodynamical framework for the Katok map, including existence and uniqueness of equilibrium measures for a family of potentials, and proves statistical properties for these measures.

Findings

Existence of equilibrium measures for all continuous potentials.

Uniqueness of equilibrium measures for the geometric t-potential when t ≠ 1.

Exponential decay of correlations and CLT for certain equilibrium measures.

Abstract

We effect thermodynamical formalism for the non-uniformly hyperbolic $C^\infty$ map of the two dimensional torus known as the Katok map. It is a slowdown of a linear Anosov map near the origin and it is a local (but not small) perturbation. We prove the existence of equilibrium measures for any continuous potential function and obtain uniqueness of equilibrium measures associated to the geometric $t$-potential $\varphi_t=-t\log |df|_{E^u(x)}|$ for any $t\in(t_0,\infty)$, $t\neq 1$ where $E^u(x)$ denotes the unstable direction. We show that $t_0$ tends to $-\infty$ as the size of the perturbation tends to zero. Finally, we establish exponential decay of correlations as well as the Central Limit Theorem for the equilibrium measures associated to $\varphi_t$ for all values of $t\in (t_0, 1)$.