Unique equilibrium states, large deviations and Lyapunov spectra for the Katok Map

TL;DR

This paper investigates the thermodynamic formalism of the Katok map, establishing existence and uniqueness of equilibrium states, deriving large deviation principles, and analyzing multifractal spectra related to Lyapunov exponents.

Contribution

It proves the existence and uniqueness of equilibrium states for certain potentials on the Katok map and analyzes their multifractal spectra, advancing understanding of non-uniform hyperbolic systems.

Findings

Existence and uniqueness of equilibrium states for specific potentials.

Derivation of level-2 large deviation principles.

Analysis of multifractal spectra for entropy and Lyapunov exponents.

Abstract

We study the thermodynamic formalism of a $C^{\infty}$ non-uniformly hyperbolic diffeomorphism on the 2-torus, known as the Katok map. We prove for a H\"older continuous potential with one additional condition, or the geometric t-potential $\varphi_t$ with $t<1$, the equilibrium state exists and is unique. We derive the level-2 large deviation principle for the equilibrium state of $\varphi_t$. We study the multifractal spectra of the Katok map for the entropy and dimension of level sets of Lyapunov exponents.