Progress in Probability

Transcription

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2 Progress in Probability Volume 46 Series Editors Thomas Liggett Charles Newman Loren Pitt

3 Fractal Geometry and Stochastics II Christoph Bandt Siegfried Graf Martina Zăhle Editors Springer Basel AG

4 Editors' addresses: Christoph Bandt Siegfried Graf Institut fiir Mathematik und Informatik Emst-Moritz-Amdt-Universitiit Fakultăt fiir Mathematik und Informatik Universităt Passau Greifswald Passau Germany Germany Martina Zăhle Mathematisches Institut Friedrich-SchiIIer-Universităt Jena Germany Mathematics Subject Classification: 60-06, 28A80 Library ofcongress, Washington D.C. Cataloging-in-Publication Data Fractal geometry aod stochastics II / Christoph Bandt, Siegfried Graf, Martina Zăhle, editors. p. cm. - (Progress in probaiity ; v. 46) Based on a conference held in GreifswaldlKoserow, Germany, Aug. 28-Sept. 2, Includes bibliographical references. ISBN ISBN (ebook) DOI / Fractals. 2. Stochastic processes. 1. Title: Fractal geometry and stochastics 2. II. Bandt, Christoph. III. Graf, Siegfried, v' Zăhle, Martina, 1950-v' Progress in probability ; 46. QA F '.742--dc Deutsche Bibiiothek Cataloging-in-Publication Data Fractal geometry aod stochastics / Christoph Bandt... ed. - Basel ; Boston; Berlin: Birkhăuser, 2000 Literaturangaben 2 (2000) (Progress in probability ; VoI. 46) ISBN This work is subject to copyright. Al! rights are reserved, whether the whole or part of the material is concemed, specifically the rights of translation, reprinting, re-use of iiiustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission ofthe copyright owner must be obtained Springer Basel AG Originally published by Birkhliuser Verlag Basel in 2000 Softcover reprint ofthe hardcover lst edition 2000 Printed on acid-free paper produced &om chlorine-free pulp. TCF 00 ISBN

5 Contents Preface... vii Introduction... ix 1. Fractal Sets and Measures 1 L. Olsen Multifractal Geometry Yuval Peres, Wilhelm Schlag, and Boris Solomyak Sixty Years of Bernoulli Convolutions Iterated Function Systems Manuel Moran Problems on Self-similar Geometry 69 Yuval Peres and Boris Solomyak Problems on Self-similar Sets and Self-affine Sets: An Update Stochastic Processes, Random Fractals John E. Hutchinson and Ludger Riischendorf Selfsimilar Fractals and Selfsimilar Random Fractals Jean-Pierre Kahane Random Coverings and Multiplicative Processes Jacques Peyriere Recent Results on Mandelbrot Multiplicative Cascades Vladas Pipiras and Murad S. Taqqu The Weierstrass-Mandelbrot Process Provides a Series Approximation to the Harmonizable Fractional Stable Motion

6 vi Contents 4. Fractals and Dynamical Systems Gerhard Keller An Ergodic Theoretic Approach to Mean Field Coupled Maps Yakov B. Pesin Entropy and Dimension Families Associated with Equilibrium Measures for Hyperbolic Dynamical Systems Harmonic Analysis on Fractals Masatoshi Fukushima On Limit Theorems for Brownian Motions on Unbounded Fractal Sets B.M. Hambly Heat Kernels and Spectral Asymptotics for some Random Sierpinski Gaskets Umberto Mosco Lagrangian Metrics and Fractal Dynamics 269 List of Participants

7 Preface The second conference on Fractal Geometry and Stochastics was held at Greifswald/Koserow, Germany from August 28 to September 2, Four years had passed after the first conference with this theme and during this period the interest in the subject had rapidly increased. More than one hundred mathematicians from twenty-two countries attended the second conference and most of them presented their newest results. Since it is impossible to collect all these contributions in a book of moderate size we decided to ask the 13 main speakers to write an account of their subject of interest. The corresponding articles are gathered in this volume. Many of them combine a sketch of the historical development with a thorough discussion of the most recent results of the fields considered. We believe that these surveys are of benefit to the readers who want to be introduced to the subject as well as to the specialists. We also think that this book reflects the main directions of research in this thriving area of mathematics. We express our gratitude to the Deutsche Forschungsgemeinschaft whose financial support enabled us to organize the conference. The Editors

8 Introduction Fractal geometry deals with geometric objects that show a high degree of irregularity on all levels of magnitude and, therefore, cannot be investigated by methods of classical geometry but, nevertheless, are interesting models for phenomena in physics, chemistry, biology, astronomy and other sciences. In mathematics a huge class of these fractal objects arises in connection with random processes. The investigation of sample paths, zero sets, and sets of multiple points for stochastic processes using Hausdorff measure and dimension as well as other types of fractal measures and dimensions as geometrical tools is a classical but still very active field of mathematical research, mainly due to the pioneering work of P. Levy and S.J. Taylor. Thus, when Mandelbrot created the term "fractal" a number of interactions between fractal geometry and stochastics did already exist. In recent years the number of research papers on the borderline between fractal geometry and probability theory has increased rapidly. Aside from the classical random sets mentioned above more and more new methods to generate random (and deterministic) fractals and fractal measures are discovered. The multifractal formalisms provide tools for the investigation of the geometric fine structure of (random) measures. The class of self-similar sets acts as a simple model for the introduction of Brownian motion and other diffusion processes on fractal objects. Thus it is possible to define a Laplace-operator on certain (random) self-similar sets and open the path for harmonic analysis on fractals. Paralleling the classical developments there are many alternative analytic ways to do harmonic analysis on fractals, for instance via Dirichlet forms or abstract harmonic spaces. Thus harmonic analysis on fractals promises to become a field of its own with many interesting connections to physics. Another class of fractal objects currently in the center of interest are the measures and sets related to dynamical systems. The present book provides a flavor of all these topics. It contains some contributions which describe the historical development of a subject from its roots to the present state of the art, while others allow a glance at the newest research development, and still others combine both features with a thorough discussion of open problems. What follows is a superficial description of the contents of this volume. In the first section fractal dimensions of certain measures and sets are studied. Lars Olsen describes the different multifractal formalisms for measures and investigates their behaviour for constructions like forming products or intersections of measures. The contribution of Yuval Peres, Wilhelm Schlag, and Boris Solomyak centers around the absolute continuity or singularity of Bernoulli convolutions.

9 x Introduction They survey the results of Erdos and Kahane and give an account of their own recent progress on the subject. The second section deals with self-similar and self-affine sets and measures generated by iterated function systems. Yuval Peres and Boris Solomyak discuss some open problems concerning dimensional properties of self-similar and selfaffine sets and give references for the relevant background. Manuel Moran considers boundaries of self-similar sets, singularity of self-similar measures with respect to Hausdorff and packing measures, and multifractal components of self-similar measures. In Section 3 different constructions of random sets and measures are described and the properties of the constructed objects are investigated. John Hutchinson and Ludger Riischendorf give a construction of random fractals via contractions which is analogous to that of self-similar sets. Jean Pierre Kahane discusses the connection between random coverings and multiplicative processes and sketches the historical development of the corresponding theory. Jacques Peyriere gives an overview of 25 years of research on Mandelbrot's multiplicative cascades which were introduced to model turbulence. Vladas Pipiras and Murad S. Taqqu show that Weierstrass's nowhere differentiable function can be modified and randomized so as to provide a series approximation to the Harmonizable Fractional Stable Motion which is a self-similar process with stationary increments and one of the many different extensions of fractional Brownian motion to the stable case. Section 4 is devoted to dynamical systems and their relationship to fractals. Gerhard Keller describes finite and infinite systems of mean field coupled maps and the limiting transition from large finite to infinite systems of this type. Yakov Pesin considers equilibrium measures corresponding to a subshift of finite type and the negative multiples of a given Holder continuous map. In particular he describes the entropy and dimension families derived from these measures and related problems. Section 5 deals with different aspects of harmonic analysis on (self-similar) fractals. Masatoshi Fukushima investigates the large time asymptotics of Brownian motion on unbounded nested fractals. Ben Hambly constructs Laplace operators on two types of randomized nested fractals using Dirichlet form techniques. Also using the theory of Dirichlet forms U mberto Mosco studies fractals as spaces of homogeneous type in the sense of abstract harmonic analysis.