Geometric pressure for multimodal maps of the interval / Feliks Przytycki, Juan Rivera-Letelier.

Format:

Book

Author/Creator:

Przytycki, Feliks, author.

Rivera-Letelier, Juan, 1975- author.

Series:

Memoirs of the American Mathematical Society ; Volume 259, Number 1246.

Memoirs of the American Mathematical Society ; Volume 259, Number 1246

Language:

English

Subjects (All):

Conformal geometry.

Mappings (Mathematics).

Riemann surfaces.

Physical Description:

1 online resource (v, 81 pages).

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2019]

Summary:

"This memoir 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 I in R into 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 pullbacks). 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"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction: The main results

1.1. Generalized multimodal maps, maximal invariant sets and related notions

1.2. Periodic orbits and basins of attraction: Bounded distortion property

1.3. Statement of Theorem A: Analytic dependence of geometric pressure on temperature, equilibria

1.4. Characterizations of geometric pressure

1.5. Non-uniformly hyperbolic interval maps

1.6. Complementary Remarks

1.7. Acknowledgements

Chapter 2. Preliminaries

2.1. Basic properties of generalized multimodal pairs

2.2. On exceptional sets

2.3. Backward stability

2.4. Weak isolation

2.5. Bounded Distortion and related notions

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

6.1. Nice sets and couples

6.2. Canonical induced map

6.3. Pressure function of the canonical induced map

Chapter 7. Analytic dependence of geometric pressure on temperature. Equilibria

7.1. The analyticity

7.2. From the induced map to the original map: Conformal measure and proof of Theorem A.2

7.3. Equilibrium states

Chapter 8. Proof of key lemma: Induced pressure

Appendix A. More on generalized multimodal maps

A.1. Darboux property approach and other observations

A.2. On topological transitivity and related notions

Appendix B. Uniqueness of equilibrium via inducing

Appendix C. Conformal pressures

Bibliography

Back Cover.

Notes:

Description based on print version record.

Includes bibliographical references.

ISBN:

1-4704-5247-2