Thermodynamics of Towers of Hyperbolic Type

We introduce a class of continuous maps f of a compact topological space X admitting inducing schemes of hyperbolic type and describe the associated tower constructions. We then establish a thermodynamic formalism, i.e., we describe a class of real-valued potential functions φ on X such that f possess a unique equilibrium measure μφ, associated to each φ, which minimizes the free energy among the measures that are liftable to the tower. We also describe some ergodic properties of equilibrium measures including decay of correlations and the central limit theorem. We then study the liftability problem and show that under some additional assumptions on the inducing scheme every measure that charges the base of the tower and has sufficiently large entropy is liftable. Our results extend those obtained in [PS05, PS08] for inducing schemes of expanding types and apply to certain multidimensional maps. Applications include obtaining the thermodynamic formalism for Young’s diffeomorphisms, the Hénon family at the first birfucation and the Katok map. In particular for the Katok map we obtain the exponential decay of correlations for equilibrium measures associated to the geometric potentials with 0 ≤ t < 1.