| Description |
1 online resource (xiv, 536 pages) : illustrations (some color) |
| Series |
Lecture notes in mathematics, 1617-9692 ; volume 2290 |
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CIRM Jean-Morlet series |
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Lecture notes in mathematics (Springer-Verlag) ; 2290. 1617-9692
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| Contents |
Intro -- Foreword -- Preface -- Contents -- Contributors -- Part I Specifications and Expansiveness -- 1 Beyond Bowen's Specification Property -- 1.1 Introduction -- 1.2 Main Ideas: Uniqueness of the Measure of Maximal Entropy -- 1.2.1 Entropy and Thermodynamic Formalism -- 1.2.2 Bowen's Original Argument: The Symbolic Case -- 1.2.2.1 The Specification Property in a Shift Space -- 1.2.2.2 The Lower Gibbs Bound as the Mechanism for Uniqueness -- 1.2.2.3 Building a Gibbs Measure -- 1.2.3 Relaxing Specification: Decompositions of the Language -- 1.2.3.1 Decompositions |
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1.2.3.2 An Example: Beta Shifts -- 1.2.3.3 Periodic Points -- 1.2.4 Beyond Shift Spaces: Expansivity in Bowen's Argument -- 1.2.4.1 Topological Entropy -- 1.2.4.2 Expansivity -- 1.2.4.3 Specification -- 1.2.4.4 Bowen's Proof Revisited -- 1.3 Non-uniform Bowen Hypotheses and Equilibrium States -- 1.3.1 Relaxing the Expansivity Hypothesis -- 1.3.2 Derived-from-Anosov Systems -- 1.3.2.1 Construction of the Mañé Example -- 1.3.2.2 Estimating the Entropy of Obstructions -- 1.3.2.3 Specification for Mañé Examples -- 1.3.3 The General Result for MMEs in Discrete-Time |
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1.3.4 Partially Hyperbolic Systems with One-Dimensional Center -- 1.3.4.1 A Small Collection of Obstructions -- 1.3.4.2 A Good Collection with Specification -- 1.3.5 Unique Equilibrium States -- 1.3.5.1 Topological Pressure -- 1.3.5.2 Regularity of the Potential Function: The Bowen Property -- 1.3.5.3 The Most General Discrete-Time Result -- 1.3.5.4 Partial Hyperbolicity -- 1.4 Geodesic Flows -- 1.4.1 Geometric Preliminaries -- 1.4.1.1 Overview -- 1.4.1.2 Surfaces -- 1.4.1.3 Invariant Foliations via Horospheres -- 1.4.1.4 Jacobi Fields and Local Construction of Stables/Unstables |
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1.4.2 Equilibrium States for Geodesic Flows -- 1.4.2.1 The General Uniqueness Result for Flows -- 1.4.2.2 Geodesic Flows in Non-positive Curvature -- 1.4.2.3 Uniqueness Can Fail Without a Pressure Gap -- 1.4.2.4 Uniqueness Given a Pressure Gap -- 1.4.2.5 Pressure and Periodic Orbits -- 1.4.2.6 Main Ideas of the Proof of Uniqueness -- 1.4.2.7 Unique MMEs for Surfaces Without Conjugate Points -- 1.4.2.8 Geodesic Flows on Metric Spaces -- 1.4.3 Kolmogorov Property for Equilibrium States -- 1.4.3.1 Moving Up the Mixing Hierarchy -- 1.4.3.2 Ledrappier's Approach -- 1.4.3.3 Decompositions for Products |
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1.4.3.4 Expansivity Issues -- 1.4.4 Knieper's Entropy Gap -- 1.4.4.1 Entropy in the Singular Set -- 1.4.4.2 Warm-Up: Shifts with Specification -- 1.4.4.3 Entropy Gap for Geodesic Flow -- 1.4.4.4 Other Applications of Pressure Production -- References -- 2 The Role of Continuity and Expansiveness on Leo and Periodic Specification Properties -- 2.1 Introduction -- 2.2 Definitions -- 2.3 Proofs -- 2.3.1 Proof of Theorem 1.2 -- 2.3.2 Proof of Theorem 1.3 -- 2.3.3 Proof of Theorem 1.4 -- 2.4 Examples -- References -- Part II Low Dimensional Dynamics and Thermodynamics Formalism |
| Summary |
This volume arose from a semester at CIRM-Luminy on "Thermodynamic Formalism: Applications to Probability, Geometry and Fractals" which brought together leading experts in the area to discuss topical problems and recent progress. It includes a number of surveys intended to make the field more accessible to younger mathematicians and scientists wishing to learn more about the area. Thermodynamic formalism has been a powerful tool in ergodic theory and dynamical system and its applications to other topics, particularly Riemannian geometry (especially in negative curvature), statistical properties of dynamical systems and fractal geometry. This work will be of value both to graduate students and more senior researchers interested in either learning about the main ideas and themes in thermodynamic formalism, and research themes which are at forefront of research in this area |
| Notes |
Online resource; title from PDF title page (SpringerLink, viewed October 5, 2021) |
| Subject |
Ergodic theory.
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Fractals.
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Thermodynamics -- Mathematics
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fractals.
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Fractales
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Termodinámica -- Matemáticas
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Teoría ergódica
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Ergodic theory
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Fractals
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Thermodynamics -- Mathematics
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Teoria ergòdica.
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Fractals.
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Termodinàmica.
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| Genre/Form |
Electronic books
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Llibres electrònics.
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| Form |
Electronic book
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| Author |
Pollicott, Mark, editor.
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Vaienti, Sandro, editor
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Centre de rencontres mathématiques de Luminy.
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| ISBN |
9783030748630 |
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3030748634 |
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