Fractal Geometry and Stochastics III

264 Pages·2004·7.32 MB·English

by  Andrzej Lasota

Progress in Probability
Volume 57
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 Umberto Mosco
Institut fUr Mathematik und Informatik Department of Physics
Emst-Moritz-Amdt-Universităt University of Rome La Sapienza
17487 Greifswald Via G. Boni 20
Germany 00162 Roma
e-mail: [email protected] 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 <http://dnb.ddb.de>.
ISBN 978-3-0348-9612-2 ISBN 978-3-0348-7891-3 (eBook)
DOI 10.1007/978-3-0348-7891-3
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© 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
987654321 www.birkhauser-science.com
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 Yimin Xiao
Additive Levy Processes: Capacity and Hausdorff Dimension ................ 151
vi Contents
4. Fractal Analysis in Euclidean Space
Hans Thebel
The Fractal Laplacian and Multifractal Quantities .......................... 173
Jenny Harrison
Geometric Representations of Currents and Distributions .................... 193
Maria Agostina Vivaldi
Variational Principles and Transmission Conditions for Fractal Layers ....... 205
5. Harmonic Analysis on Fractals
Takashi Kumagai
Function Spaces and Stochastic Processes on Fractals ....................... 221
Al! Jonsson
A Dirichlet Form on the Sierpinski Gasket,
Related Function Spaces, and Traces ........................................ 235
Alexander Teplyaev
Spectral Zeta Function of Symmetric Fractals ............................... 245
Preface
The conference on Fractal Geometry and Stochastics which took place at Friedrich
roda, 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 e-mail 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 Forschungsgemein
schaft for their financial support which was essential for organizing the conference.
The Editors
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 mul
tifractal 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 dy
namical systems, each working with specific methods, and Chapter 2 will high
light 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 the
orem 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 turbu
lence, as well as applications to covering and percolation problems. Recent exciting
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 mea
sure and the properties of the Courant function are discussed. In a more abstract
approach, J. Harrison considers fractal measures as currents and proves a geomet
ric 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.
Partl
Fractal Sets and Measures