Thermodynamic Formalism and Applications to Dimension Theory
This self-contained monograph presents a unified exposition of the thermodynamic formalism and some of its main extensions, with emphasis on the relation to dimension theory and multifractal analysis of dynamical systems. In particular, the book considers
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Series Editors Hyman Bass Joseph Oesterlé Yuri Tschinkel Alan Weinstein
Luis Barreira
Thermodynamic Formalism and Applications to Dimension Theory
Luis Barreira Departamento de Matemática Instituto Superior Técnico 1049-001 Lisboa Portugal [email protected]
2010 Mathematical Subject Classification: 37D35, 37D20, 37D25, 37C45 ISBN 978-3-0348-0205-5 e-ISBN 978-3-0348-0206-2 DOI 10.1007/978-3-0348-0206-2 Library of Congress Control Number: 2011936516 © Springer Basel AG 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media www.birkhauser-science.com
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Contents Preface
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1 Introduction 1.1 Thermodynamic formalism and dimension theory . . . . 1.1.1 Classical thermodynamic formalism . . . . . . . 1.1.2 Dimension theory and multifractal analysis . . . 1.1.3 Attractors in infinite-dimensional spaces . . . . . 1.2 Extensions and applications . . . . . . . . . . . . . . . . 1.2.1 Nonadditive formalism and dimension estimates 1.2.2 Subadditive formalism and entropy spectra . . . 1.2.3 Almost additive formalism and Gibbs measures . 1.3 Contents of the book . . . . . . . . . . . . . . . . . . . . 1.4 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Dimension theory . . . . . . . . . . . . . . . . . . 1.4.2 Ergodic theory . . . . . . . . . . . . . . . . . . .
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I Classical Thermodynamic Formalism
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2 Thermodynamic Formalism: Basic Notions 2.1 Topological pressure . . . . . . . . . . 2.2 Equivalent definitions of pressure . . . 2.3 Variational principle . . . . . . . . . . 2.4 Equilibrium measures . . . . . . . . .
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3 The 3.1 3.2 3.3 3.4
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Case of Symbolic Dynamics Topological pressure . . . . Two-sided sequences . . . . Equilibrium measures . . . Gibbs measures . . . . . . .
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Contents
II Nonadditive Thermodynamic Formalism
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4 Nonadditive Thermodynamic Formalism 4.1 Nonadditive topological pressure . . . . . . . . . . 4.2 Properties of the pressure . . . . . . . . . . . . . . 4.2.1 Dependence on the data . . . . . . . . . . . 4.2.2 Characterizations of the capacity pressures 4.2.3 Subadditive sequences . . . . . . . . . . . . 4.3
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