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Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot
Edited by: Michel L. Lapidus, University of California, Riverside, CA, and Machiel van Frankenhuijsen, Utah Valley State College, Orem, UT
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Proceedings of Symposia in Pure Mathematics
2004; 1091 pp; hardcover
Volume: 72
ISBN-10: 0-8218-3292-1
ISBN-13: 978-0-8218-3292-9
List Price: US$186
Member Price: US$148.80
Order Code: PSPUM/72
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This volume offers an excellent selection of cutting-edge articles about fractal geometry, covering the great breadth of mathematics and related areas touched by this subject. Included are rich survey articles and fine expository papers. The high-quality contributions to the volume by well-known researchers--including two articles by Mandelbrot--provide a solid cross-section of recent research representing the richness and variety of contemporary advances in and around fractal geometry.

In demonstrating the vitality and diversity of the field, this book will motivate further investigation into the many open problems and inspire future research directions. It is suitable for graduate students and researchers interested in fractal geometry and its applications.

This is a two-part volume. Part 1 covers analysis, number theory, and dynamical systems; Part 2, multifractals, probability and statistical mechanics, and applications.

Readership

Graduate students and research mathematicians interested in fractal geometry and its applications.

Table of Contents

Part I
  • M. L. Lapidus -- Fractal geometry and applications-An introduction to this volume
  • J. Barral and S. Jaffard -- Cherche Livre... et plus si affinité/Looking for a book...and more, if affinity
  • M. Berry -- Benefiting from fractals
  • M.-O. Coppens -- Benoit Mandelbrot, wizard of science
  • R. L. Devaney -- Mandelbrot's vision for mathematics
  • M. M. Dodson -- Benoit Mandelbrot and York
  • B. Duplantier -- Nul n'entre ici s'il n'est géomètre/Let no one ignorant of geometry enter here
  • M. L. Frame -- A decade of working with a maverick
  • M. Frantz -- Breakfast with Mandelbrot
  • J.-P. Kahane -- Old memories
  • D. B. Mumford -- My encounters with Benoit Mandelbrot
  • L. Nottale -- Fractal geometry and the foundations of physics
  • B. Sapoval -- Is randomness partially tamed by fractals?
  • J. E. Taylor -- On knowing Benoit Mandelbrot
Analysis
  • M. M. France -- Reflections, ripples and fractals
  • M. Frantz -- Lacunarity, Minkowski content, and self-similar sets in \(\mathbb{R}\)
  • F. Morgan -- Fractals and geometric measure theory: Friends and foes
  • H. Furstenberg and Y. Katznelson -- Eigenmeasures, equidistribution, and the multiplicity of \(\beta\)-expansions
  • A. Kameyama -- Distances on topological self-similar sets
  • A. Teplyaev -- Energy and laplacian on the Sierpiński gasket
  • C. Sabot -- Electrical networks, symplectic reductions, and application to the renormalization map of self-similar lattices
  • B. Solomyak -- Notes on Bernoulli convolutions
Number theory
  • T. Hilberdink -- Some connections between Bernoulli convolutions and analytic number theory
  • S. Jaffard -- On Davenport expansions
  • M. M. Dodson and S. Kristensen -- Hausdorff dimension and diophantine approximation
  • M. L. Lapidus and M. van Frankenhuijsen -- Fractality, self-similarity and complex dimensions
Dynamical systems
  • B. Kahng -- The invariant fractals of symplectic piecewise affine elliptic dynamics
  • S. Crovisier -- Almost sure rotation number of circle endomorphisms
  • V. Baladi -- Kneading determinants and transfer operators in higher dimensions
  • V. Afraimovich, L. Ramírez, and E. Ugalde -- The spectrum of dimensions for Poincaré recurrences for nonuniformly hyperbolic geometric constructions
  • M. Comerford -- A survey of results in random iteration
  • D. Schleicher -- On fibers and local connectivity of Mandelbrot and multibrot sets
Part II
Multifractals
  • J. Barral and B. B. Mandelbrot -- Introduction to infinite products of independent random functions (Random multiplicative multifractal measures, part I)
  • J. Barral and B. B. Mandelbrot -- Non-degeneracy, moments, dimension, and multifractal analysis for random multiplicative measures (Random multiplicative multifractal measures, part II)
  • J. Barral -- Techniques for the study of infinite products of independent random functions (Random multiplicative multifractal measures, part III)
  • S. P. Jaffard -- Wavelet techniques in multifractal analysis
  • J. L. Véhel and S. Seuret -- The 2-microlocal formalism
  • J. Peyrière -- A vectorial multifractal formalism
Probability and statistical mechanics
  • B. M. Hambly and T. Kumagai -- Heat kernel estimates for symmetric random walks on a class of fractal graphs and stability under rough isometries
  • Y. Xiao -- Random fractals and Markov processes
  • G. F. Lawler, O. Schramm, and W. Werner -- On the scaling limit of planar self-avoiding walk
  • B. Duplantier -- Conformal fractal geometry & boundary quantum gravity
Applications
  • A. Desolneux, B. Sapoval, and A. Baldassarri -- Self-organized percolation power laws with and without fractal geometry in the etching of random solids
  • M.-O. Coppens -- Nature inspired chemical engineering-Learning from the fractal geometry of nature in sustainable chemical engineering
  • F. K. Musgrave -- Fractal forgeries of nature
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