Thermodynamics of some non-uniformly hyperbolic attractors

Abstract

We study thermodynamic formalism for certain dissipative maps, that is maps with non-uniformly hyperbolic attractors, which are obtained from uniformly hyperbolic systems by the slow down procedure. Namely, starting with a hyperbolic local diffeomorphism with an attractor , one slows down trajectories in a small neighborhood of a hyperbolic fixed point obtaining a nonuniformly hyperbolic diffeomorphism with a topological attractor . We establish the existence of equilibrium measures for the family of geometric t-potentials defined by . We identify equilibrium measures for t = 1. Under additional restrictions we prove the existence of t0 < 0 such that the equilibrium measures are unique for every that belongs to the interval . We show that for the equilibrium measures have exponential decay of correlations and satisfy the central limit theorem. Our results apply to any diffeomorphism which is a small perturbation of the classical Smale–Williams Solenoid, thus providing examples of nonuniformly hyperbolic dissipative maps on a three-dimensional manifold for which one can build thermodynamic formalism.