Fractal Geometry and Stochastics III
Fractal geometry is used to model complicated natural and technical phenomena in various disciplines like physics, biology, finance, and medicine. Since most convincing models contain an element of randomness, stochastics enters the area in a natural way.
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Series Editors Thomas Liggett Charles Newman Loren Pitt Sidney I. Resnick
Fractal Geometry and Stochastics III Christoph Bandt Umberto Mosco Martina Zahle Editors
Springer Basel AG
Editors: Christoph Bandt Institut fUr Mathematik und Informatik Emst-Moritz-Amdt-Universităt 17487 Greifswald Germany e-mail: [email protected]
Umberto Mosco Department of Physics University of Rome La Sapienza Via G. Boni 20 00162 Roma Italy e-mail: [email protected]
Martina Ziihle Mathematisches Institut Friedrich-Schiller-Universităt 07740 Jena Germany e-mail: [email protected]
2000 Mathematics Subject Classification: 20HI0, 26E25, 28Axx, 37C45, 46E35, 53C65, 60Gxx, 60Jxx
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA
Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at . ISBN 978-3-0348-9612-2 ISBN 978-3-0348-7891-3 (eBook) DOI 10.1007/978-3-0348-7891-3
This work is subject to copyright. AH rights are reserved, whether the whole or part of the material is concemed, specificaHy the rights of translation, reprinting, re-use of illustrations, 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. © 2004 Springer Basel AG Originally published by Birkhăuser Verlag Basel in 2004 Softcover reprint of the hardcover lst edition 2004 Printed on acid-free paper produced from chlorine-free pulp. rCF 00 ISBN 978-3-0348-9612-2
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Contents
Preface ...................................................................... vii Introduction ................................................................. ix 1. Fractal Sets and Measures
Andrzej Lasota, J6ze! Myjak and Tomasz Szarek Markov Operators and Semifractals ........................................... 3 Jacques Levy Whel and Claude Tricot On Various Multifractal Spectra ............................................. 23 Zhi-Ying Wen One-Dimensional Moran Sets and the Spectrum of Schrodinger Operators .... 43
2. Fractals and Dynamical Systems Alben M. Fisher Small-scale Structure via Flows .............................................. 59 Karoly Simon Hausdorff Dimension of Hyperbolic Attractors in IR3 ......................... 79 Bernd O. Stratmann The Exponent of Convergence of Kleinian Groups; on a Theorem of Bishop and Jones ........................................... 93 Amiran Ambroladze and Jory Schmeling Lyapunov Exponents Are not Rigid with Respect to Arithmetic Subsequences ................................... 109
3. Stochastic Processes and Random fractals Ai Hua Fan Some Topics in the Theory of Multiplicative Chaos ......................... 119 Peter Moners Intersection Exponents and the Multifractal Spectrum for Measures on Brownian Paths ............................................ 135 Davar Khoshnevisan and Yi
