Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry
The theory of random dynamical systems originated from stochastic differential equations. It is intended to provide a framework and techniques to describe and analyze the evolution of dynamical systems when the input and output data are known only approxi
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For further volumes: http://www.springer.com/series/304
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Volker Mayer Bartlomiej Skorulski Mariusz Urbanski
Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry
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Volker Mayer Universit´e Lille 1 D´epartement de Math´ematiques 59655 Villeneuve d’Ascq France [email protected]
Mariusz Urbanski University of North Texas Department of Mathematics Denton, TX 76203-1430 USA [email protected]
Bartlomiej Skorulski Universidad Catolica del Norte Departamento de Matematicas Avenida Angamos 0610 Antofagasta Chile [email protected]
ISBN 978-3-642-23649-5 e-ISBN 978-3-642-23650-1 DOI 10.1007/978-3-642-23650-1 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2011940286 Mathematics Subject Classification (2010): 37-XX c Springer-Verlag Berlin Heidelberg 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
In this book we introduce measurable expanding random systems, develop the thermodynamical formalism and establish, in particular, exponential decay of correlations and analyticity of the expected pressure although the spectral gap property does not hold. This theory is then used to investigate fractal properties of conformal random systems. We prove a Bowen’s formula and develop the multifractal formalism of the Gibbs states. Depending on the behavior of the Birkhoff sums of the pressure function we get a natural classifications of the systems into two classes: quasi-deterministic systems which share many properties of deterministic ones and essential random systems which are rather generic and never bi-Lipschitz equivalent to deterministic systems. We show in the essential case that the Hausdorff measure vanishes which refutes a conjecture of Bogensch¨utz and Ochs. We finally give applications of our results to various specific conformal random systems and positively answer a question of Br¨uck and B¨uger concerning the Hausdorff dimension of randomJulia sets.
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Acknowledgements
The second author was supported by FON
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