Fractal Geometry and Analysis

1. Front Matter

   Pages i-xv

   PDF

2. Applications of dynamical systems theory to fractals — a study of cookie-cutter Cantor sets

   • Tim Bedford

   Pages 1-44

3. Complex dynamics: an informal discussion

   • Paul Blanchard, Amy Chiu

   Pages 45-98

4. Substitutions, branching processes and fractal sets

   • F. Michel Dekking

   Pages 99-119

5. Interpolation fractale

   • Serge Dubuc

   Pages 121-220

6. Dimensions — their determination and properties

   • Kenneth J. Falconer

   Pages 221-254

7. Topological aspects of self-similar sets and singular functions

   • Masayoshi Hata

   Pages 255-276

8. Produits de poids aléatoires indépendants et applications

   • Jean-Pierre Kahane

   Pages 277-324

9. The Planck constant of a curve

   • Michel Mendès France

   Pages 325-366

10. Rectifiable and fractal sets

    • Claude Tricot

    Pages 367-403

11. Iterated function systems: theory, applications and the inverse problem

    • Edward R. Vrscay

    Pages 405-468

12. Back Matter

    Pages 469-472

    PDF

This ASI- which was also the 28th session of the Seminaire de mathematiques superieures of the Universite de Montreal - was devoted to Fractal Geometry and Analysis. The present volume is the fruit of the work of this Advanced Study Institute. We were fortunate to have with us Prof. Benoit Mandelbrot - the creator of numerous concepts in Fractal Geometry - who gave a series of lectures on multifractals, iteration of analytic functions, and various kinds of fractal stochastic processes. Different foundational contributions for Fractal Geometry like measure theory, dy­ namical systems, iteration theory, branching processes are recognized. The geometry of fractal sets and the analytical tools used to investigate them provide a unifying theme of this book. The main topics that are covered are then as follows. Dimension Theory. Many definitions of fractional dimension have been proposed, all of which coincide on "regular" objects, but often take different values for a given fractal set. There is ample discussion on piecewise estimates yielding actual values for the most common dimensions (Hausdorff, box-counting and packing dimensions). The dimension theory is mainly discussed by Mendes-France, Bedford, Falconer, Tricot and Rata. Construction of fractal sets. Scale in variance is a fundamental property of fractal sets.

• Branching process
• calculus
• dynamical systems
• dynamische Systeme
• geometry
• systems theory

• Département de mathématiques et de statistique, Université de Montréal, Montréal, Canada

  Jacques Bélair, Serge Dubuc