Conformal Fractals
Ergodic Theory Methods

A one-stop introduction to the methods of ergodic theory applied to holomorphic iteration that is ideal for graduate courses.

Feliks Przytycki (Author), Mariusz Urba?ski (Author)

9780521438001, Cambridge University Press

Paperback, published 6 May 2010

366 pages, 26 b/w illus. 90 exercises
22.8 x 15.2 x 1.8 cm, 0.52 kg

'This is an interesting and substantial book which makes a valuable contribution to the theory of iterations of expanding and non-uniformly expanding holomorphic maps, an area with a long tradition as well as a lot of current activities … An extensive bibliography and a useful index complete this essential reference in ergodic theory for conformal fractals.' Zentralblatt MATH

This is a one-stop introduction to the methods of ergodic theory applied to holomorphic iteration. The authors begin with introductory chapters presenting the necessary tools from ergodic theory thermodynamical formalism, and then focus on recent developments in the field of 1-dimensional holomorphic iterations and underlying fractal sets, from the point of view of geometric measure theory and rigidity. Detailed proofs are included. Developed from university courses taught by the authors, this book is ideal for graduate students. Researchers will also find it a valuable source of reference to a large and rapidly expanding field. It eases the reader into the subject and provides a vital springboard for those beginning their own research. Many helpful exercises are also included to aid understanding of the material presented and the authors provide links to further reading and related areas of research.

Introduction
Basic examples and definitions
1. Measure preserving endomorphisms
2. Compact metric spaces
3. Distance expanding maps
4. Thermodynamical formalism
5. Expanding repellers in manifolds and in the Riemann sphere, preliminaries
6. Cantor repellers in the line, Sullivan's scaling function, application in Feigenbaum universality
7. Fractal dimensions
8. Conformal expanding repellers
9. Sullivan's classification of conformal expanding repellers
10. Holomorphic maps with invariant probability measures of positive Lyapunov exponent
11. Conformal measures
References
Index.

Subject Areas: Fractal geometry [PBMX], Differential calculus & equations [PBKJ], Calculus & mathematical analysis [PBK]