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.

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.

A one-stop introduction to the methods of ergodic theory applied to holomorphic iteration. 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 area of research.

Feliks Przytycki is a Professor in the Institute of Mathematics at the Polish Academy of Sciences.
Mariusz Urbanski is a Professor in the Department of Mathematics at the University of North Texas.