Fractal Geometry and Stochastics

Kenneth J. Falconer (auth.), Christoph Bandt, Siegfried Graf, Martina Zähle (eds.)9780817652630, 0-8176-5263-9, 3764352639

Fractal geometry is a new and promising field for researchers from different disciplines such as mathematics, physics, chemistry, biology and medicine. It is used to model complicated natural and technical phenomena. The most convincing models contain an element of randomness so that the combination of fractal geometry and stochastics arises in between these two fields. It contains contributions by outstanding mathematicians and is meant to highlight the principal directions of research in the area. The contributors were the main speakers attending the conference “Fractal Geometry and Stochastics” held at Finsterbergen, Germany, in June 1994. This was the first international conference ever to be held on the topic. The book is addressed to mathematicians and other scientists who are interested in the mathematical theory concerning: • Fractal sets and measures • Iterated function systems • Random fractals • Fractals and dynamical systems, and • Harmonic analysis on fractals. The reader will be introduced to the most recent results in these subjects. Researchers and graduate students alike will benefit from the clear expositions.

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Table of contents :
Front Matter….Pages i-xi
Front Matter….Pages 1-1
Probabilistic Methods in Fractal Geometry….Pages 3-13
Measures of Fractal Lacunarity: Minkowski Content and Alternatives….Pages 15-42
Tangent Measures, Densities, and Singular Integrals….Pages 43-52
Front Matter….Pages 53-53
Self-Similarity, L p -Spectrum and Multifractal Formalism….Pages 55-90
Infinite Iterated Function Systems: Theory and Applications….Pages 91-110
Front Matter….Pages 111-111
Aspects of the Fractal Percolation Process….Pages 113-143
Fractals via Random Iterated Function Systems and Random Geometric Constructions….Pages 145-164
Front Matter….Pages 165-165
Remarks on Weak Limit Laws for Fractal Sets….Pages 167-178
A Rigidity Theorem in Complex Dynamics….Pages 179-187
Front Matter….Pages 189-189
Estimates on the Spectrum of Fractals Arising from Affine Iterations….Pages 191-219
Laplacians on Self-Similar Sets and their Spectral Distributions….Pages 221-238
Back Matter….Pages 239-248