**CBMS Regional Conference Series in Mathematics**

Volume: 120;
2014;
69 pp;
Softcover

MSC: Primary 28; 37;
Secondary 30; 47

Print ISBN: 978-1-4704-1034-6

Product Code: CBMS/120

List Price: $32.00

Individual Member Price: $25.60

**Electronic ISBN: 978-1-4704-1854-0
Product Code: CBMS/120.E**

List Price: $32.00

Individual Member Price: $25.60

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#### Supplemental Materials

# Ergodic Theory and Fractal Geometry

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*Hillel Furstenberg*

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.

—Endre Szemerédi, Rényi
Institute of Mathematics, Budapest

Fractal geometry represents a radical departure from classical
geometry, which focuses on smooth objects that “straighten
out” under magnification. Fractals, which take their name from
the shape of fractured objects, can be characterized as retaining
their lack of smoothness under magnification. The properties of
fractals come to light under repeated magnification, which we refer to
informally as “zooming in”. This zooming-in process has
its parallels in dynamics, and the varying “scenery”
corresponds to the evolution of dynamical variables.

The present monograph focuses on applications of one branch of
dynamics—ergodic theory—to the geometry of fractals. Much
attention is given to the all-important notion of fractal dimension,
which is shown to be intimately related to the study of ergodic
averages. It has been long known that dynamical systems serve as a
rich source of fractal examples. The primary goal in this monograph is
to demonstrate how the minute structure of fractals is unfolded when
seen in the light of related dynamics.

A co-publication of the AMS and CBMS.

#### Table of Contents

# Table of Contents

## Ergodic Theory and Fractal Geometry

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Preface ix10 free
- Introduction to fractals 112 free
- Dimension 1122 free
- Trees and fractals 1526
- Invariant sets 2132
- Probability trees 2334
- Galleries 2738
- Probability trees revisited 3142
- Elements of ergodic theory 3344
- Galleries of trees 3546
- General remarks on Markov systems 3748
- Markov operator 𝒯 and measure preserving transformation 𝒯 3950
- Probability trees and galleries 4354
- Ergodic theorem and the proof of the main theorem 4758
- An application: The 𝑘-lane property 5162
- Dimension and energy 5364
- Dimension conservation 5566
- Ergodic theorem for sequences of functions 5768
- Dimension conservation for homogeneous fractals: The main steps in the proof 5970
- Verifying the conditions of the ergodic theorem for sequences of functions 6576
- Bibliography 6778
- Index 6980 free
- Back Cover Back Cover182

#### Readership

Graduate students and research mathematicians interested in fractal geometry and ergodic theory.