Thermodynamics of the Katok Map
Yakov Pesin, Samuel Senti, Ke Zhang

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 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 -potential for any , where denotes the unstable direction. We show that tends to 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 for all values of .
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