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London Mathematical Society Lecture Note Series : Conformal Fractals: Ergodic Theory Methods Series Number 371 - Feliks Przytycki

London Mathematical Society Lecture Note Series

Conformal Fractals: Ergodic Theory Methods Series Number 371

Paperback Published: 6th May 2010
ISBN: 9780521438001
Number Of Pages: 366

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This is a comprehensive and self-contained introduction to the methods of ergodic theory applied to holomorphic iteration. The authors begin with introductory chapters presenting the necessary tools from ergodic theory and thermodynamical formalism, and then focus on recent developments in the field of 1-dimensional holomorphic iterations and underlying fractal sets, from the point of view of geometric measure theory and rigidity Detailed proofs are included.

Developed from university courses taught by the authors, this book is ideal for graduate students. Researchers will also find it a valuable source of reference to a large and rapidly expanding field, It eases the reader into the subject and provides a vital springboard for those beginning their own research. The authors also include many helpful exercises to aid understanding of the material presented, and provide links to further reading and related areas of research.

'This is an interesting and substantial book which makes a valuable contribution to the theory of iterations of expanding and non-uniformly expanding holomorphic maps, an area with a long tradition as well as a lot of current activities ... An extensive bibliography and a useful index complete this essential reference in ergodic theory for conformal fractals.' Zentralblatt MATH
"This book provides a broad and self-contained introduction to the theory of uniformly and non-uniformly expanding holomorphic maps. This book is a valuable resource for fundamental results and a comprehensive state of the art reference for recent developments in the field." Katrin Gelfert, Mathematical Reviews
"Conformal Fractals is packed with classical gems with proofs provided. The authors are experts on extending the subject in many of the important directions it has taken the past several decades, especially the move from uniformly hyperbolic maps to expansive maps, which includes many rational maps of the sphere. This is an interesting text that could be used for a year-long graduate course in ergodic theory..." Jane Hawkins, Bulletin of the American Mathematical Society

Introductionp. 1
Basic examples and definitionsp. 8
Measure-preserving endomorphismsp. 17
Measure spaces and the Martingale Theoremp. 17
Measure-preserving endomorphisms; ergodicityp. 20
Entropy of partitionp. 26
Entropy of an endomorphismp. 29
Shannon-McMillan-Breiman Theoremp. 33
Lebesgue spaces, measurable partitions and canonical systems of conditional measuresp. 36
Rokhlin natural extensionp. 41
Generalized entropy; convergence theoremsp. 46
Countable-to-one maps, Jacobian and entropy of endomorphismsp. 50
Mixing propertiesp. 53
Probability laws and Bernoulli propertyp. 56
Exercisesp. 60
Bibliographical notesp. 65
Ergodic theory on compact metric spacesp. 67
Invariant measures for continuous mappingsp. 67
Topological pressure and topological entropyp. 75
Pressure on compact metric spacesp. 79
Variational Principlep. 81
Equilibrium states and expansive mapsp. 85
Topological pressure as a function on the Banach space of continuous functions; the issue of uniqueness of equilibrium statesp. 89
Exercisesp. 98
Bibliographical notesp. 100
Distance-expanding mapsp. 102
Distance-expanding open maps: basic propertiesp. 103
Shadowing of pseudo-orbitsp. 105
Spectral decomposition; mixing propertiesp. 107
Hölder continuous functionsp. 113
Markov partitions and symbolic representationp. 118
Expansive maps are expanding in some metricp. 125
Exercisesp. 127
Bibliographical notesp. 129
Thermodynamical formalismp. 131
Gibbs measures: introductory remarksp. 131
Transfer operator and its conjugate; measures with prescribed Jacobiansp. 134
Iteration of the transfer operator; existence of invariant Gibbs measuresp. 141
Convergence of Ln; mixing properties of Gibbs measuresp. 144
More on almost periodic operatorsp. 151
Uniqueness of equilibrium statesp. 153
Probability laws and 2(u, v)p. 157
Exercisesp. 162
Bibliographical notesp. 165
Expanding repellers in manifolds and in the Riemann sphere: preliminariesp. 166
Basic propertiesp. 167
Complex dimension one; bounded distortion and other techniquesp. 172
Transfer operator for conformal expanding repeller with harmonic potentialp. 175
Analytic dependence of transfer operator on potential functionp. 179
Exercisesp. 184
Bibliographical notesp. 184
Cantor repellers in the line; Sullivan's scaling function; application in Feigenbaum universalityp. 185
C1+-equivalencep. 186
Scaling function: C1+-extension of the shift mapp. 192
Higher smoothnessp. 196
Scaling function and smoothness; Cantor set valued scaling functionp. 200
Cantor set generating familiesp. 204
Quadratic-like maps of the interval; an application to Feigenbaum's universalityp. 206
Exercisesp. 213
Bibliographical notesp. 214
Fractal dimensionsp. 216
Outer measuresp. 216
Hausdorff measuresp. 219
Packing measuresp. 222
Dimensionsp. 223
Besicovitch Covering Theorem; Vitali Theorem and density pointsp. 226
Frostman-type lemmasp. 231
Bibliographical notesp. 235
Conformal expanding repellersp. 236
Pressure function and dimensionp. 237
Multifractal analysis of Gibbs statep. 245
Fluctuations for Gibbs measuresp. 256
Boundary behaviour of the Riemann mapp. 260
Harmonic measure; 'fractal vs. analytic' dichotomyp. 264
Pressure versus integral means of the Riemann mapp. 273
Geometric examples: snowflake and Carleson's domainsp. 275
Exercisesp. 279
Bibliographical notesp. 282
Sullivan's classification of conformal expanding repellersp. 284
Equivalent notions of linearityp. 284
Rigidity of non-linear CERsp. 288
Bibliographical notesp. 294
Holomorphic maps with invariant probability measures of positive Lyapunov exponentp. 295
Ruelle's inequalityp. 295
Pesin's theoryp. 297
Mañé's partitionp. 300
Volume Lemma and the formula HD () = h(f)/(f)p. 302
Pressure-like definition of the functional h + ∫ dp. 305
Katok's theory: hyperbolic sets, periodic points, and pressurep. 308
Exercisesp. 313
Bibliographical notesp. 313
Conformal measuresp. 314
General notion of conformal measuresp. 314
Sullivan's conformal measures and dynamical dimension: Ip. 320
Sullivan's conformal measures and dynamical dimension: IIp. 322
Pesin's formulap. 327
More about geometric pressure and dimensionsp. 329
Bibilographical notesp. 335
Referencesp. 336
Indexp. 349
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780521438001
ISBN-10: 0521438004
Series: London Mathematical Society Lecture Note Series
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 366
Published: 6th May 2010
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 22.8 x 15.4  x 2.2
Weight (kg): 0.51

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