Fractal Geometry and Stochastics

1. Front Matter

   Pages i-xi

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2. Fractal Sets and Measures

   1. Front Matter

      Pages 1-1

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   2. Probabilistic Methods in Fractal Geometry

      • Kenneth J. Falconer

      Pages 3-13

   3. Measures of Fractal Lacunarity: Minkowski Content and Alternatives

      • Benoit B. Mandelbrot

      Pages 15-42

   4. Tangent Measures, Densities, and Singular Integrals

      • Pertti Mattila

      Pages 43-52

3. Iterated Function Systems

   1. Front Matter

      Pages 53-53

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   2. Self-Similarity, L ^p-Spectrum and Multifractal Formalism

      • Ka-Sing Lau

      Pages 55-90

   3. Infinite Iterated Function Systems: Theory and Applications

      • R. Daniel Mauldin

      Pages 91-110

4. Random Fractals

   1. Front Matter

      Pages 111-111

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   2. Aspects of the Fractal Percolation Process

      • Lincoln Chayes

      Pages 113-143

   3. Fractals via Random Iterated Function Systems and Random Geometric Constructions

      • Yuri Kifer

      Pages 145-164

5. Fractals and Dynamical Systems

   1. Front Matter

      Pages 165-165

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   2. Remarks on Weak Limit Laws for Fractal Sets

      • Manfred Denker

      Pages 167-178

   3. A Rigidity Theorem in Complex Dynamics

      • Mariusz Urbański, Alexander Volberg

      Pages 179-187

6. Harmonic Analysis on Fractals

   1. Front Matter

      Pages 189-189

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   2. Estimates on the Spectrum of Fractals Arising from Affine Iterations

      • Palle E. T. Jorgensen, Steen Pedersen

      Pages 191-219

   3. Laplacians on Self-Similar Sets and their Spectral Distributions

      • Jun Kigami

      Pages 221-238

7. Back Matter

   Pages 239-248

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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.

• Peak
• dynamical systems
• dynamische Systeme
• fractal
• fractal geometry
• geometry
• mathematics
• medicine
• model
• stochastics

• Fachbereich Mathematik und Informatik, Ernst-Moritz-Arndt-Universität, Greifswald, Deutschland

  Christoph Bandt

• Fakultät für Mathematik und Informatik, Universität Passau, Passau, Deutschland

  Siegfried Graf

• Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität, Jena, Deutschland

  Martina Zähle

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