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Equilibrium measures for maps with inducing schemes. (English) Zbl 1159.37007

The paper introduces a class of continuous maps \(f\) of a compact topological space \(I\) admitting inducing schemes and describes the tower constructions associated with them. For these maps, a thermodynamic formalism is established, i.e., a class of real-valued potential functions \(\varphi\) on \(I\), which admit a unique equilibrium measure \(\mu_\varphi\) minimizing the free energy for a certain class of invariant measures, is described. The paper describes also certain ergodic properties of equilibrium measures, including decay of correlation and the Central Limit Theorem. The results obtained are applied to certain interval maps with critical points and/or singularities (including some unimodal and multimodal maps) and to potential functions of the form \(\varphi_t=-t\log|df|\) with \(t\in (t_0,t_1)\) for some \(t_0<1<t_1\). In the particular case of \(S\)-unimodal maps, it is shown that one can choose \(t_0<0\) and that the class of measures under consideration consists of all invariant Borel probability measures.

MSC:

37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37E05 Dynamical systems involving maps of the interval
37E10 Dynamical systems involving maps of the circle
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