Thermodynamics of the Katok map

We effect the thermodynamical formalism for the non-uniformly hyperbolic C map of the two-dimensional torus known as the Katok map [Katok. Bernoulli diffeomorphisms on surfaces. Ann. of Math. (2) 110(3) 1979, 529-547]. It is a slow-down 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 φ_t =-t log |df|E^u(x)| for any t ∞ (t₀, ∞), t ≠ 1 where E^u(x) denotes the unstable direction. We show that t₀ tends to-∞ as the domain of the perturbation shrinks to zero. Finally, we establish exponential decay of correlations as well as the central limit theorem for the equilibrium measures associated to φ_t for all values of t ∞ (t₀, 1).