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 φt =-t log |df|Eu(x)| for any t ∞ (t0, ∞), t ≠ 1 where Eu(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 φt for all values of t ∞ (t0, 1).
All Science Journal Classification (ASJC) codes
- Applied Mathematics