Progress in Probability

Transcription

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2 Progress in Probability Volume 57 Series Editors Thomas Liggett Charles Newman Loren Pitt Sidney I. Resnick

3 Fractal Geometry and Stochastics III Christoph Bandt Umberto Mosco Martina Zahle Editors Springer Basel AG

4 Editors: Christoph Bandt Institut fur Mathematik und Informatik Emst-Moritz-Amdt-Universităt Greifswald Germany Martina Ziihle Mathematisches Institut Friedrich-Schiller-Universităt Jena Germany Umberto Mosco Department of Physics University of Rome La Sapienza Via G. Boni Roma Italy 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 ISBN (ebook) DOI / 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 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

5 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 Zhi-Ying Wen One-Dimensional Moran Sets and the Spectrum of Schrodinger Operators Fractals and Dynamical Systems Alben M. Fisher Small-scale Structure via Flows Karoly Simon Hausdorff Dimension of Hyperbolic Attractors in IR Bernd O. Stratmann The Exponent of Convergence of Kleinian Groups; on a Theorem of Bishop and Jones Amiran Ambroladze and Jory Schmeling Lyapunov Exponents Are not Rigid with Respect to Arithmetic Subsequences Stochastic Processes and Random fractals Ai Hua Fan Some Topics in the Theory of Multiplicative Chaos Peter Moners Intersection Exponents and the Multifractal Spectrum for Measures on Brownian Paths Davar Khoshnevisan and Yimin Xiao Additive Levy Processes: Capacity and Hausdorff Dimension

6 vi Contents 4. Fractal Analysis in Euclidean Space Hans Thebel The Fractal Laplacian and Multifractal Quantities Jenny Harrison Geometric Representations of Currents and Distributions Maria Agostina Vivaldi Variational Principles and Transmission Conditions for Fractal Layers Harmonic Analysis on Fractals Takashi Kumagai Function Spaces and Stochastic Processes on Fractals Al! Jonsson A Dirichlet Form on the Sierpinski Gasket, Related Function Spaces, and Traces Alexander Teplyaev Spectral Zeta Function of Symmetric Fractals

7 Preface The conference on Fractal Geometry and Stochastics which took place at Friedrichroda, Germany, from March 17 to 22, 2003, was the third in a series. The previous conferences were held in 1994 and 1998, and their main lectures were published by Birkhauser, Progress in Probability, Volumes 37 and 46. During recent years the interest in the subject has still been growing, and more profound results were obtained. The size of the conference also increased, with 119 participants from all over the world. Abstracts of most of the contributions, a list of addresses and some photos can be found on the web page WNW.minet. uni -jena. de/fgs3/. For this volume, we have asked only the 16 main speakers of the conference to work out their contributions carefully, for a large audience. In order to present new facets of the subject, and new experts working in the field, we decided not to include any authors of the 1994 and 1998 volumes. The papers were chosen to represent the main directions of contemporary research in the area. Most of them are surveys, written in a comprehensible style. A few also contain original results. We hope that non-experts as well as specialists will benefit from the presentations in this book. We would like to express our gratitude to the Deutsche Forschungsgemeinschaft for their financial support which was essential for organizing the conference. The Editors

8 Introduction Fractal geometry is known for its applications. Irregular phenomena in physics and material science, in biology and medicine, in economy and finance can be described by fractal dimensions and multifractal spectra. In recent years it turned out that fractals are also at the core of several deep and intricate problems of analysis, mathematical physics and probability theory, as for instance renormalization in dynamical systems and conformal invariance of percolation. Thus some concepts of fractal geometry have been successfully exploited in purely mathematical research. On the other hand, fractal geometry benefitted from the use of analytical and probabilistic methods. Among others, random walks and diffusion processes on fractals are studied by means of Laplace operators and Dirichlet forms. This development is documented in the present book. Fractal analysis and the use of function spaces will be central themes. Let us briefly review the contents of this volume. Already in the first chapter on general fractal sets and measures, linear spaces play a role - as a tool and also as a subject where fractals come in. A. Lasota, J. Myjak and T. Szarek put the well-known construction of self-similar measures into the general setting of Markov operators and derive conclusions for the large class of semifractals. In the paper by Z.-Y. Wen it is shown that the spectra of certain Schrodinger operators are Cantor sets with general self-similarity properties. Methods for determining the dimensions of such sets are worked out. J. Levy-Vehel and C. Tricot study multifractal spectra arising from Hausdorff dimension and from the principle of large deviations in probability. They define modified spectra which can be estimated more easily in an experimental framework. Most fractals come from dynamical systems. There are many classes of dynamical systems, each working with specific methods, and Chapter 2 will highlight some important directions. K. Simon clarifies the difficulties of the structure of hyperbolic at tractors of differentiable maps and shows how to estimate their Hausdorff dimension. In the case of Kleinian groups acting on hyperbolic space, the Hausdorff dimension of the limit set is better understood. A fundamental theorem says that it coincides with the exponent of convergence of the group. B. Stratmann presents a simplified proof of this basic result. A.M. Fisher describes an approach from ergodic theory to Kleinian limit sets as well as to many other fractals. He studies the magnification flow which arises when we zoom in at a certain point of the fractal. Self-similarity of the fractal implies ergodicity of the flow, and the entropy of the flow equals the Hausdorff dimension of the fractal. A special random walk constructed by A. Ambroladze and J. Schmeling shows that a dynamical system can have positive Lyapunov exponent while the exponent is zero on each arithmetic subsequence. Random fractals and stochastic processes are studied in Chapter 3. A.H. Fan describes the theory of multiplicative chaos which originated in a model of turbulence, as well as applications to covering and percolation problems. Recent exciting

9 x Introduction work on the intersection exponents for Brownian paths is reviewed by P. Morters, who points out the relations to the multifractal structure of certain measures on the paths. D. Koshnevisan and Y. Xiao study the potential theory of additive Levy processes. They find the Hausdorff dimension of the range of the process and of the sets of multiple points. There are two chapters on fractal analysis. In Chapter 4, the fractals are subsets of Euclidean space, and function spaces on the surrounding JRn are used. H. Triebel studies the fractal Laplacian of a Radon measure in the plane, and its eigenfunction corresponding to the largest eigenvalue, called Courant function. Connections between the multifractal and Besov characteristics of the given measure and the properties of the Courant function are discussed. In a more abstract approach, J. Harrison considers fractal measures as currents and proves a geometric representation theorem for currents. M.A. Vivaldi investigates transmission problems with fractal layers. Chapter 5 deals with the analysis on fractal structures without reference to the surrounding space. T. Kumagai studies Dirichlet forms which connect function spaces with diffusion processes on fractals. Heat kernel estimates are given in a very general setting. A. Jonsson considers a special Dirichlet form on the Sierpinski gasket function spaces, and the connections between the corresponding Lipschitz and Besov spaces. The paper by A. Teplyaev also starts with the gasket, observing that the spectrum of the Laplace operator is obtained from a quadratic function. Spectral zeta functions of similar examples are discussed.