Thermodynamics of the Katok Map

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 ${\phi }_t=-t\log |df{|}_{E^u(x)}|$ for any $t\in (t_0,\infty )$, $t\ne 1$ 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 ${\phi }_t$ for all values of $t\in (t_0,1)$.

31 pages to appear in Ergod. Th. & Dy.nam. Sys

References in corpus (1)

• Invariant manifolds and equilibrium states for non-uniformly hyperbolic horseshoes

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