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Geometric pressure for multimodal maps of the interval

About this Title

Feliks Przytycki and Juan Rivera-Letelier

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)
Published electronically: May 1, 2019

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Table of Contents


  • Chapter 1. Introduction: The main results
  • Chapter 2. Preliminaries
  • Chapter 3. Non-uniformly hyperbolic interval maps
  • Chapter 4. Equivalence of the definitions of geometric pressure
  • Chapter 5. Pressure on periodic orbits
  • Chapter 6. Nice inducing schemes
  • Chapter 7. Analytic dependence of geometric pressure on temperature. Equilibria
  • Chapter 8. Proof of key lemma: Induced pressure
  • Appendix A. More on generalized multimodal maps
  • Appendix B. Uniqueness of equilibrium via inducing
  • Appendix C. Conformal pressures


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 of a finite union of compact intervals in into with non-flat critical points, such that on its maximal forward invariant set the map is topologically transitive and has positive topological entropy. We prove that several notions of non-uniform hyperbolicity of are equivalent (including uniform hyperbolicity on periodic orbits, TCE & all periodic orbits in hyperbolic repelling, Lyapunov hyperbolicity, and exponential shrinking of pull-backs). We prove that several definitions of geometric pressure , that is pressure for the map and the potential , 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 are hyperbolic repelling, the function is real analytic for between the "condensation" and "freezing" parameters and that for each such there exists unique equilibrium (and conformal) measure satisfying strong statistical properties.

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