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Geometric pressure for multimodal maps of the interval
About this Title
Feliks Przytycki, Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00656 Warszawa, Poland. and Juan Rivera-Letelier, University of Rochester, UR Mathematics, 811 Hylan Building, University of Rochester, RC Box 270138, Rochester, NY 14627, U.S.A.
Publication: Memoirs of the American Mathematical Society
Publication Year:
2019; Volume 259, Number 1246
ISBNs: 978-1-4704-3567-7 (print); 978-1-4704-5247-6 (online)
DOI: https://doi.org/10.1090/memo/1246
Published electronically: May 1, 2019
MSC: Primary 37E05, 37D25, 37D35.
Table of Contents
Chapters
- 1. Introduction: The main results
- 2. Preliminaries
- 3. Non-uniformly hyperbolic interval maps
- 4. Equivalence of the definitions of geometric pressure
- 5. Pressure on periodic orbits
- 6. Nice inducing schemes
- 7. Analytic dependence of geometric pressure on temperature. Equilibria
- 8. Proof of key lemma: Induced pressure
- A. More on generalized multimodal maps
- B. Uniqueness of equilibrium via inducing
- C. Conformal pressures
Abstract
This paper is an interval dynamics counterpart of three theories founded earlier by the authors, S. Smirnov and others in the setting of the iteration of rational maps on the Riemann sphere: the equivalence of several notions of non-uniform hyperbolicity, Geometric Pressure, and Nice Inducing Schemes methods leading to results in thermodynamical formalism. We work in a setting of generalized multimodal maps, that is smooth maps $f$ of a finite union of compact intervals $\widehat {I}$ in $\mathbb {R}$ into $\mathbb {R}$ with non-flat critical points, such that on its maximal forward invariant set $K$ the map $f$ is topologically transitive and has positive topological entropy. We prove that several notions of non-uniform hyperbolicity of $f|_K$ are equivalent (including uniform hyperbolicity on periodic orbits, TCE & all periodic orbits in $K$ hyperbolic repelling, Lyapunov hyperbolicity, and exponential shrinking of pull-backs). We prove that several definitions of geometric pressure $P(t)$, that is pressure for the map $f|_K$ and the potential $-t\log |f’|$, give the same value (including pressure on periodic orbits, "tree" pressure, variational pressures and conformal pressure). Finally we prove that, provided all periodic orbits in $K$ are hyperbolic repelling, the function $P(t)$ is real analytic for $t$ between the "condensation" and "freezing" parameters and that for each such $t$ there exists unique equilibrium (and conformal) measure satisfying strong statistical properties.- Lluís Alsedà, Jaume Llibre, and MichałMisiurewicz, Combinatorial dynamics and entropy in dimension one, 2nd ed., Advanced Series in Nonlinear Dynamics, vol. 5, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. MR 1807264, DOI 10.1142/4205
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