Equilibrium measures for maps with inducing schemes

We introducea class of continuousmaps f of a compact topologi- cal space I admitting inducing schemes and describe the tower constructions associated with them. We then establish a thermodynamicformalism, i.e., de- scribe a class of real-valued potential functions ϕ on I, which admit a unique equilibrium measure ¹ ϕ minimizing the free energy for a certain class of in- variant measures. We also describe ergodic properties of equilibrium mea- sures, including decay of correlation and the Central Limit Theorem. Our re- sults apply to certain maps of the interval with critical points and/or singular- ities (including some unimodal and multimodal maps) and to potential func- tions ϕt =−t log|d f | with t ∈(t0,t1) for some t0 < 1< t1. In theparticularcase of S-unimodal maps we show that one can choose t0 < 0 and that the class of measures under consideration consists of all invariant Borel probability mea- sures.