Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II: Fractals in Applied Mathematics

384 Pages·2013·8.49 MB·English

by  David Carfi

601
Fractal Geometry and Dynamical Systems
in Pure and Applied Mathematics II:
Fractals in Applied Mathematics
PISRS 2011 International Conference on Analysis, Fractal Geometry,
Dynamical Systems and Economics
November 2011: Messina, Sicily, Italy
AMS Special Session on Fractal Geometry in Pure and Applied Mathematics:
in Memory of Benoît Mandelbrot
January 2012: Boston, Massachusetts
AMS Special Session on Geometry and Analysis on Fractal Spaces
March 2012: Honolulu, Hawaii
David Carfì
Michel L. Lapidus
Erin P. J. Pearse
Machiel van Frankenhuijsen
Editors
AmericanMathematicalSociety
Fractal Geometry and Dynamical Systems
in Pure and Applied Mathematics II:
Fractals in Applied Mathematics
PISRS 2011 International Conference on Analysis,Fractal Geometry,
Dynamical Systems and Economics
November 2011: Messina,Sicily,Italy
AMS SpecialSessiononFractal Geometry in Pure and AppliedMathematics:
in Memory of BenoîtMandelbrot
January 2012: Boston, Massachusetts
AMS SpecialSessionon Geometry and Analysis on FractalSpaces
March 2012: Honolulu,Hawaii
David Carfì
Michel L. Lapidus
Erin P. J. Pearse
Machiel van Frankenhuijsen
Editors

601
Fractal Geometry and Dynamical Systems
in Pure and Applied Mathematics II:
Fractals in Applied Mathematics
PISRS 2011 International Conference on Analysis,Fractal Geometry,
Dynamical Systems and Economics
November 2011: Messina,Sicily,Italy
AMS SpecialSessiononFractal Geometry in Pure and AppliedMathematics:
in Memory of BenoîtMandelbrot
January 2012: Boston, Massachusetts
AMS SpecialSessionon Geometry and Analysis on FractalSpaces
March 2012: Honolulu,Hawaii
David Carfì
Michel L. Lapidus
Erin P. J. Pearse
Machiel van Frankenhuijsen
Editors
AmericanMathematicalSociety
Providence,RhodeIsland
EDITORIAL COMMITTEE
Dennis DeTurck, Managing Editor
Michael Loss Kailash Misra Martin J. Strauss
2010 Mathematics Subject Classification. Primary 28A80, 37D50, 37F10, 58J35, 58J50,
58J65, 60J45, 60K35, 81Q35, 91G80.
Library of Congress Cataloging-in-Publication Data
Fractal geometry and dynamical systems in pure and applied mathematics / David Carf`ı,
MichelL.Lapidus,ErinP.J.Pearse,MachielvanFrankenhuijsen,editors.
volumescm. –(Contemporarymathematics;volumes600,601)
PISRS2011,FirstInternationalConference: Analysis,FractalGeometry,DynamicalSystemsand
Economics,November8–12,2011,Messina,Sicily,Italy.
AMS Special Session, in memory of Benoˆıt Mandelbrot: Fractal Geometryin Pure and Applied
Mathematics,January4–7,2012,Boston,MA.
AMSSpecialSession: GeometryandAnalysisonFractalSpaces,March3–4,2012,Honolulu,HI.
Includesbibliographicalreferences.
ISBN978-0-8218-9147-6(alk. paper: v. I)–ISBN978-0-8218-9148-3(alk. paper: v. II)
1. Fractals–Congresses. I.Carf`ı,David,1971–II.Lapidus,MichelL.(MichelLaurent),1956–
III.Pearse,ErinP.J.,1975–IV.Frankenhuijsen,Machielvan,1967–V.Mandelbrot,BenoˆıtB.
QC20.7.F73F7152013 2013013894
514(cid:2).742–dc23
ContemporaryMathematicsISSN:0271-4132(print);ISSN:1098-3627(online)
DOI:http://dx.doi.org/10.1090/conm/601
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Contents
Preface vii
Statistical Mechanics and Quantum Fields on Fractals
Eric Akkermans 1
Spectral Algebra of the Chernov and Bogoslovsky Finsler Metric Tensors
Vladimir Balan 23
Local Multifractal Analysis
Julien Barral, Arnaud Durand, St´ephane Jaffard,
and St´ephane Seuret 31
Extreme Risk and Fractal Regularity in Finance
Laurent E. Calvet and Adlai J. Fisher 65
An Algorithm for Dynamical Games with Fractal-Like Trajectories
David Carf´ı and Angela Ricciardello 95
The Landscape of Anderson Localization in a Disordered Medium
Marcel Filoche and Svitlana Mayboroda 113
Zeta Functions for Infinite Graphs and Functional Equations
Daniele Guido and Tommaso Isola 123
Vector Analysis on Fractals and Applications
Michael Hinz and Alexander Teplyaev 147
Non-Regularly Varying and Non-Periodic Oscillation of the On-Diagonal
Heat Kernels on Self-Similar Fractals
Naotaka Kajino 165
Lattice Effects in the Scaling Limit of the Two-Dimensional
Self-Avoiding Walk
Tom Kennedy and Gregory F. Lawler 195
The Casimir Effect on Laakso Spaces
Robert Kesler and Benjamin Steinhurst 211
The Decimation Method for Laplacians on Fractals: Spectra and Complex
Dynamics
Nishu Lal and Michel L. Lapidus 227
The Current State of Fractal Billiards
Michel L. Lapidus and Robert G. Niemeyer 251
v
vi CONTENTS
Long-Range Dependence and the Rank of Decompositions
C´eline L´evy-Leduc and Murad S. Taqqu 289
Hitting Probabilities of the Random Covering Sets
Bing Li, Narn-Rueih Shieh, and Yimin Xiao 307
Fractal Oscillations Near the Domain Boundary of Radially Symmetric
Solutions of p-Laplace Equations
Yu¯ki Naito, Mervan Paˇsic´, Satoshi Tanaka,
and Darko Zˇubrinic´ 325
Applications of the Contraction Mapping Principle
John R. Quinn 345
Economics and Psychology. Perfect Rationality versus Bounded Rationality
Daniele Schiliro` 359
Preface
The Contemporary Mathematics volume
Fractal Geometry and Dynamical Systems in Pure and Applied
Mathematics II: Fractals in Applied Mathematics
contains papers from talks given at three conferences held in 2011–2012, following
thepassingofBenoˆıtMandelbrot(widelyregardedasthefatheroffractalgeometry)
in October of 2010. These meetings are described in chronological order below.
On the occasion of the 2011 Anassilaos International Research Prize in Math-
ematics, awarded to Michel L. Lapidus (University of California, Riverside), the
Permanent International Session of Research Seminars (PISRS) held its first Inter-
national Meeting
PISRS 2011: Analysis, Fractal Geometry, Dynamical Systems
and Economics.
The conference was held on November 8–12, 2011, at the University of Messina in
Sicily, Italy, and was attended by experts in the fields of Fractal Geometry, Dy-
namical Systems, Number Theory, Noncommutative Geometry, Mathematical and
Theoretical Physics, as well as Economics. In addition to approximately 40 expe-
rienced researchers participating, the conference included more than 150 students,
professors and experts following and attending the meeting. The Award Cere-
monyforMichelLapidustookplaceinReggioCalabriaonSaturday, November12.
The Scientific Committee of PISRS includes over 50 professors and scholars from
more than 25 outstanding universities around the world. It has several branches,
including Applied Functional Analysis; Biomathematics; Decision and Game The-
ory; Differential, Fractal and Noncommutative Geometry; Mathematical Methods
of Economics, Finance and Quantum Mechanics; Mathematical Physics and Dy-
namical Systems. The Chairman of PISRS is David Carf`ı.
The 2012 AMS/MAA/SIAM Joint Mathematics National Meeting, held in
BostoninJanuary2012,includedanAMSSpecialSessionon“FractalGeometryin
Pure and Applied Mathematics” in memory of Benoˆıt Mandelbrot. Its organizers
were Michel Lapidus, Erin Pearse and Machiel van Frankenhuijsen. In five ses-
sions (including sessions comprised of primarily applied topics), researchers from
around the world presented their work in various areas of fractal mathematics.
An entire session was devoted to the applications to Physics, Biology, Engineering
and Computer Science. During one of the breaks, an experiment was performed
which demonstrated the capabilities of fractal antennas. Many speakers described
ways in which their work was influenced by the work of Benoˆıt Mandelbrot, and a
vii
viii PREFACE
special dinner was organized in his honor. Several talks were attended by Aliette
Mandelbrot, Benoˆıt’s widow, who also gave a short but touching speech.
The Spring 2012 Meeting of the AMS Western Section, held in Honolulu,
Hawaii, at the University of Hawaii at Manoa, included a Special Session on “Ge-
ometry and Analysis on Fractal Spaces”. Its organizers were Michel Lapidus, Lu˜’
Hu`ng, John Rock and Machiel van Frankenhuijsen. In four sessions, researchers
from around the world presented their work in various areas of fractal mathemat-
ics.
This is a collection of papers on fractal geometry and dynamical systems in
applied mathematics and the applications to other sciences. It features articles
discussing a variety of connections between these subjects and other fields of sci-
ence, including physics, engineering, computer science, technology, economics and
finance, as well as of mathematics (including probability theory in relation with
statistical physics and heat kernel estimates, geometric measure theory, partial dif-
ferential equations in relation with condensed matter physics, global analysis on
nonsmooth spaces, the theory of billiards, harmonic analysis and spectral theory).
These proceedings were conceived as a means of collecting some of the most
recentdevelopmentsinthisactiveareaofresearch,andalsotobringtogetherseveral
survey and research expository articles, as a means of introducing new researchers
and graduate students to the forefront of the field. The present volume focuses on
themore appliedaspectsofthefield, including theapplications offractalgeometry
and dynamical systems to other sciences. Its companion volume, entitled Fractal
GeometryandDynamicalSystemsinPureandAppliedMathematicsI andsubtitled
Fractals in Pure Mathematics, focuseson the more mathematical aspectsof fractal
geometry and dynamical systems.
David Carf`ı,
Michel L. Lapidus,
Erin P. J. Pearse, and
Machiel van Frankenhuijsen.
March 2013
Acknowledgements: The editors wish to acknowledge the support of the Na-
tionalScienceFoundation(viaM.L.Lapidus’NSFgrantsDMS-0707524andDMS-
1107750) towards the preparation of these proceedings and especially towards the
travel and/or stay of several of the participants in the three conferences that gave
rise to these proceedings.
ContemporaryMathematics
Volume601,2013
http://dx.doi.org/10.1090/conm/601/11962
Statistical Mechanics and Quantum Fields on Fractals
Eric Akkermans
This paperis dedicatedto the memoryof B. Mandelbrot.
Abstract. Fractalsdefineanewandinterestingrealmforadiscussionofba-
sicphenomenainquantumfieldtheoryandstatisticalmechanics. Thisinterest
resultsfromspecificpropertiesoffractals,e.g.,theirdilatationsymmetryand
thecorrespondingabsenceofFouriermodedecomposition. Moreover,theex-
istence of a set of distinct dimensions characterizing the physical properties
(spatialorspectral)offractalsmakethemausefultestinggroundfordimen-
sionalitydependentphysicalproblems. Thispaperaddressesspecificproblems
including thebehavior ofthe heatkerneland spectralzetafunctions onfrac-
talsandtheirimportanceintheexpressionofspectralpropertiesinquantum
physics. Finally, we apply these results to specific problems such as thermo-
dynamicsofquantumradiationbyafractalblackbody.
1. Introduction
The interest in the behavior of fractals (a word coined by B. Mandelbrot in
the 1970’s [1] but without well agreed definition) goes back to the study by math-
ematicians of strange objects hardly defined by their topology such as the Koch
curve or the Sierpinski gasket. These objects are described by continuous but not
differentiablefunctions. Ataboutthesametime,probabilists(Levy,Wiener,Doob,
Ito, Kolmogorov) and physicists (Smoluchowski, Einstein, Perrin and Langevin to
name a few) have been working to give a basis to the theory of brownian motion,
yet another example of fractal object. More recently, physicists have recognized
theubiquitouscharacteroffractalsinalmost all fieldof physics, includingcomplex
condensedmatter([2]-[8]), phasetransitions[9,10], turbulence[11],quantumfield
theory ([12]-[16]) and aspects of stochastic processes [7,17,18]. An important ef-
fortspanovermorethantwodecadesledtonewideasandconceptstocharacterize
fractals. Notions of self-similarity, iterative maps, fixed points and the identifica-
tion of distinct fractal dimensions to characterize basic physical properties have
been instrumental in the understanding of these objects.
Since the early 1980’s, mathematicians have opened new important directions
by being able to define properly brownian motion on some classes of fractals ([19]-
[21]) and Laplacian operators on these structures [23,24]. Progress along these
2010 Mathematics Subject Classification. Primary 81Q35, 28A80, 60G18, 82B10, 37F25,
37F10.
Key wordsand phrases. Statisticalmechanics,Quantumfieldtheory,Fractals.
TheauthorwassupportedinpartbytheIsraelScienceFoundationGrantNo.924/09.
(cid:3)c2013 American Mathematical Society
1