Meromorphic Dynamics: Abstract Ergodic Theory, Geometry, Graph Directed Markov Systems, and Conformal Measures

Janina Kotus; Mariusz Urbański

doi:10.1017/9781009215930

This text, the first of two volumes, provides a comprehensive and self-contained introduction to a wide range of fundamental results from ergodic theory and geometric measure theory. Topics covered include: finite and infinite abstract ergodic theory, Young's towers, measure-theoretic Kolmogorov-Sinai entropy, thermodynamics formalism, geometric function theory, various kinds of conformal measures, conformal graph directed Markov systems and iterated functions systems, semi-local dynamics of analytic functions, and nice sets. Many examples are included, along with detailed explanations of essential concepts and full proofs, in what is sure to be an indispensable reference for both researchers and graduate students.

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Contents

Frontmatter
pp i-iv
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Dedication
pp v-vi
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Contents
pp vii-x
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Contents of Volume II
pp xi-xiv
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Preface
pp xv-xix
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Acknowledgments
pp xx-xx
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Introduction
pp xxi-xxvi
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Part I - Ergodic Theory and Geometric Measures
pp 1-2
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1 - Geometric Measure Theory
pp 3-54
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2 - Invariant Measures: Finite and Infinite
pp 55-101
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3 - Probability (Finite) Invariant Measures: Basic Properties and Existence
pp 102-121
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4 - Probability (Finite) Invariant Measures: Finer Properties
pp 122-133
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5 - Infinite Invariant Measures: Finer Properties
pp 134-159
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6 - Measure-Theoretic Entropy
pp 160-191
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7 - Thermodynamic Formalism
pp 192-246
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Part II - Complex Analysis, Conformal Measures, and Graph Directed Markov Systems
pp 247-248
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8 - Selected Topics from Complex Analysis
pp 249-315
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9 - Invariant Measures for Holomorphic Maps f in A(X) or in Aw(X)
pp 316-341
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10 - Sullivan Conformal Measures for Holomorphic Maps f in A(X) and in Aw(X)
pp 342-377
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11 - Graph Directed Markov Systems
pp 378-432
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12 - Nice Sets for Analytic Maps
pp 433-468
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References
pp 469-477
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Index of Symbols
pp 478-479
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Subject Index
pp 480-482
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Meromorphic Dynamics

Abstract Ergodic Theory, Geometry, Graph Directed Markov Systems, and Conformal Measures

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Contents

Frontmatter
pp i-iv

Dedication
pp v-vi

Contents
pp vii-x

Contents of Volume II
pp xi-xiv

Preface
pp xv-xix

Acknowledgments
pp xx-xx

Introduction
pp xxi-xxvi

Part I - Ergodic Theory and Geometric Measures
pp 1-2

1 - Geometric Measure Theory
pp 3-54

2 - Invariant Measures: Finite and Infinite
pp 55-101

3 - Probability (Finite) Invariant Measures: Basic Properties and Existence
pp 102-121

4 - Probability (Finite) Invariant Measures: Finer Properties
pp 122-133

5 - Infinite Invariant Measures: Finer Properties
pp 134-159

6 - Measure-Theoretic Entropy
pp 160-191

7 - Thermodynamic Formalism
pp 192-246

Part II - Complex Analysis, Conformal Measures, and Graph Directed Markov Systems
pp 247-248

8 - Selected Topics from Complex Analysis
pp 249-315

9 - Invariant Measures for Holomorphic Maps f in A(X) or in Aw(X)
pp 316-341

10 - Sullivan Conformal Measures for Holomorphic Maps f in A(X) and in Aw(X)
pp 342-377

11 - Graph Directed Markov Systems
pp 378-432

12 - Nice Sets for Analytic Maps
pp 433-468

References
pp 469-477

Index of Symbols
pp 478-479

Subject Index
pp 480-482

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Meromorphic Dynamics

Abstract Ergodic Theory, Geometry, Graph Directed Markov Systems, and Conformal Measures

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