Thermodynamics of some non-uniformly hyperbolic attractors

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 $f:U\to M$ with an attractor $\Lambda$, one slows down trajectories in a small neighborhood of a hyperbolic fixed point $p\in \Lambda$ obtaining a nonuniformly hyperbolic diffeomorphism $g:U\to M$ with a topological attractor ${\Lambda }_g$. We establish the existence of equilibrium measures for the family of geometric t-potentials defined by ${\phi }_t\left(x\right):=-t\log \left|D{g}_{\left|E^u\right(x)}\right|$. We identify equilibrium measures for t  =  1. Under additional restrictions we prove the existence of t ₀  <  0 such that the equilibrium measures are unique for every $t\ne 1$ that belongs to the interval $(t_0,\infty )$. We show that for $t\in (t_0,1)$ 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. (paper)