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36,56 €Cover 1
Title page 4
Contents 8
Preface 10
Introduction to fractals 12
Dimension 22
Trees and fractals26
Invariant sets32
Probability trees34
Galleries38
Probability trees revisited42
Elements of ergodic theory44
Galleries of trees46
General remarks on Markov systems48
Markov operator ?? and measure preserving transformation ??50
Probability trees and galleries54
Ergodic theorem and the proof of the main theorem58
An application: The ??-lane property62
Dimension and energy64
Dimension conservation66
Ergodic theorem for sequences of functions68
Dimension conservation for homogeneous fractals: The main steps in the proof70
Verifying the conditions of the ergodic theorem for sequences of functions76
Bibliography78
Index 80
Back Cover82
A co-publication of the AMS and CBMS
Fractals are beautiful and complex geometric objects. Their study, pioneered by Benoît Mandelbrot, is of interest in mathematics, physics and computer science. Their inherent structure, based on their self-similarity, makes the study of their geometry amenable to dynamical approaches. In this book, a theory along these lines is developed by Hillel Furstenberg, one of the foremost experts in ergodic theory, leading to deep results connecting fractal geometry, multiple recurrence, and Ramsey theory. In particular, the notions of fractal dimension and self-similarity are interpreted in terms of ergodic averages and periodicity of classical dynamics; moreover, the methods have deep implications in combinatorics. The exposition is well-structured and clearly written, suitable for graduate students as well as for young researchers with basic familiarity in analysis and probability theory.
Author
Hillel Furstenberg: The Hebrew University of Jerusalem, Jerusalem, Israel
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