Equilibrium measures for maps with inducing schemes

Abstract

Abstract. We introduce a class of continuous maps f of a compact topological space I admitting inducing schemes and describe the tower constructions associated with them. We then establish a thermodynamic formalism, i.e., describe a class of real-valued potential functions ϕ on I, which admit a unique equilibrium measure µ ϕ minimizing the free energy for a certain class of invariant measures. We also describe ergodic properties of equilibrium measures including decay of correlation and the central limit theorem. Our results apply to certain maps of the interval with critical points and/or singularities (including some unimodal and multimodal maps) and to potential functions ϕ t = −t log |df | with t ∈ (t 0 , t 1 ) for some t 0 < 1 < t 1 . In the particular case of Sunimodal maps we show that one can choose t 0 < 0 and that the class of measures under consideration consists of all invariant Borel probability measures.

Thermodynamics of inducing schemes and liftability of measures

“…We describe the class of systems admitting inducing schemes, which were introduced in [19,18]. Let f : I → I be a continuous map of a compact topological space I, S a countable collection of disjoint Borel subsets of I, and τ : S → N a positive integer-valued function.…”

“…The set X is forward invariant under f . In view of (H2), the induced map F : W → W is conjugate to the one-sided Bernoulli shift σ on a countable set of states S. More precisely, this means the following (see [19,18]). Define the coding map h :…”

“…Following [19,18] we describe a class of potential functions ϕ : I → R, which admit unique equilibrium measures, i.e. for which the supremum (1) is achieved with respect to a certain class of invariant measures.…”

“…The induced map F may not be the first return map, however, Abramov's formula, connecting the entropies of F and f , and Kac's formula, connecting the integrals of ϕ and ϕ, still hold (see [19,27], for related results see also Keller [15]).…”

“…In [19,18], Pesin and Senti developed thermodynamical formalisms for general systems admitting tower constructions. In particular, they described a class of potential functions for which equilibrium measures exist and are unique; see Section 3 for detailed definitions and results.…”