Journal of Modern Dynamics (JMD)

Equilibrium measures for maps with inducing schemes

Pages: 397 - 430, Issue 3, July 2008      doi:10.3934/jmd.2008.2.397

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Yakov Pesin - Department of Mathematics, McAllister Building, Pennsylvania State University, University Park, PA 16802, United States (email)
Samuel Senti - Instituto de Matemática, Universidade Federal do Rio de Janeiro, C.P. 68 530, CEP 21945-970, R.J., Brazil (email)

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, \ie describe 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. 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 $\varphi_t=-t\log|df|$ with $t\in(t_0, t_1)$ for some $t_0<1 < t_1$. In the particular case of $S$-unimodal maps we show that one can choose $t_0<0$ and that the class of measures under consideration consists of all invariant Borel probability measures.

Keywords:  Thermodynamic formalism, equilibrium meassures, inducing schemes, towers, liftability.
Mathematics Subject Classification:  Primary: 37D25, 37D35, Secondary: 37E05, 37E10.

Received: May 2007;      Revised: April 2008;      Available Online: April 2008.