Softcover ISBN:  9781470410346 
Product Code:  CBMS/120 
69 pp 
List Price:  $34.00 
MAA Member Price:  $30.60 
AMS Member Price:  $27.20 
Electronic ISBN:  9781470418540 
Product Code:  CBMS/120.E 
69 pp 
List Price:  $32.00 
MAA Member Price:  $28.80 
AMS Member Price:  $25.60 

Book DetailsCBMS Regional Conference Series in MathematicsVolume: 120; 2014MSC: Primary 28; 37; Secondary 30; 47;
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 zoomingin 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 allimportant 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 copublication of the AMS and CBMS.
ReadershipGraduate students and research mathematicians interested in fractal geometry and ergodic theory.

Table of Contents

Chapters

1. Introduction to fractals

2. Dimension

3. Trees and fractals

4. Invariant sets

5. Probability trees

6. Galleries

7. Probability trees revisited

8. Elements of ergodic theory

9. Galleries of trees

10. General remarks on Markov systems

11. Markov operator $\mathcal {T}$ and measure preserving transformation $T$

12. Probability trees and galleries

13. Ergodic theorem and the proof of the main theorem

14. An application: The $k$lane property

15. Dimension and energy

16. Dimension conservation

17. Ergodic theorem for sequences of functions

18. Dimension conservation for homogeneous fractals: The main steps in the proof

19. Verifying the conditions of the ergodic theorem for sequences of functions


Additional Material

Reviews

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 selfsimilarity, 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 selfsimilarity are interpreted in terms of ergodic averages and periodicity of classical dynamics; moreover, the methods have deep implications in combinatorics. The exposition is wellstructured 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


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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 zoomingin 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 allimportant 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 copublication of the AMS and CBMS.
Graduate students and research mathematicians interested in fractal geometry and ergodic theory.

Chapters

1. Introduction to fractals

2. Dimension

3. Trees and fractals

4. Invariant sets

5. Probability trees

6. Galleries

7. Probability trees revisited

8. Elements of ergodic theory

9. Galleries of trees

10. General remarks on Markov systems

11. Markov operator $\mathcal {T}$ and measure preserving transformation $T$

12. Probability trees and galleries

13. Ergodic theorem and the proof of the main theorem

14. An application: The $k$lane property

15. Dimension and energy

16. Dimension conservation

17. Ergodic theorem for sequences of functions

18. Dimension conservation for homogeneous fractals: The main steps in the proof

19. Verifying the conditions of the ergodic theorem for sequences of functions

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 selfsimilarity, 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 selfsimilarity are interpreted in terms of ergodic averages and periodicity of classical dynamics; moreover, the methods have deep implications in combinatorics. The exposition is wellstructured 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