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

Abstract

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

The K-Property for Some Unique Equilibrium States in Flows and Homeomorphisms

“…where B∞ is obtained from the λ-decomposition in [30]. Consequently, by Proposition 4.7, this pressure gap holds for all potentials sufficiently close to constant, and so, for any such potential which has the Bowen property on this decomposition, the unique equilibrium state has the K-property.…”

“…Not every application of [11] has been a λ-decomposition, but many are able to be studied using this theory. For more references, see [3,6,8,9,30]. We make use of them in this paper as they behave well for the necessary pressure estimates, and they also induce natural decompositions in the product space via λ(x, y) = λ(x)λ(y), both of which allow us to make use of Theorem 3.3.…”

“…However, there are other settings in which this result can be applied. For instance, work of Climenhaga, Fisher, and Thompson on Mañé diffeomorphisms [8] uses a decomposition similar to one-sided λ-decompositions, and work of Wang on equilibrium states for the Katok map [30] also uses the setup of λ-decompositions. In what follows, we will show how this work can be applied in these settings as well to obtain the K-property.…”

“…Consequently, by Proposition 4.7, this pressure gap holds for all potentials sufficiently close to constant, and so, for any such potential which has the Bowen property on this decomposition, the unique equilibrium state has the K-property. Using [30,Proposition 6.3], this applies to all Hölder continuous potentials sufficiently close to constant. 6.2.…”

Thermodynamics of smooth models of pseudo–Anosov homeomorphisms

“…† For the Katok map, it is shown in[21] that for sufficiently small values of the parameters α > 0 and r > 0, the Katok map has a unique equilibrium measure μ t corresponding to the geometric potential ϕ t for all values of t < 1.…”