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Ergodic Theory and Fractal Geometry

Hillel Furstenberg The Hebrew University of Jerusalem, Jerusalem, Israel
A co-publication of the AMS and CBMS
Available Formats:
Softcover ISBN: 978-1-4704-1034-6
Product Code: CBMS/120
List Price: $34.00 MAA Member Price:$30.60
AMS Member Price: $27.20 Electronic ISBN: 978-1-4704-1854-0 Product Code: CBMS/120.E List Price:$32.00
MAA Member Price: $28.80 AMS Member Price:$25.60
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $51.00 MAA Member Price:$45.90
AMS Member Price: $40.80 Click above image for expanded view Ergodic Theory and Fractal Geometry Hillel Furstenberg The Hebrew University of Jerusalem, Jerusalem, Israel A co-publication of the AMS and CBMS Available Formats:  Softcover ISBN: 978-1-4704-1034-6 Product Code: CBMS/120  List Price:$34.00 MAA Member Price: $30.60 AMS Member Price:$27.20
 Electronic ISBN: 978-1-4704-1854-0 Product Code: CBMS/120.E
 List Price: $32.00 MAA Member Price:$28.80 AMS Member Price: $25.60 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$51.00 MAA Member Price: $45.90 AMS Member Price:$40.80
• Book Details

CBMS Regional Conference Series in Mathematics
Volume: 1202014; 69 pp
MSC: 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 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.

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

• 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 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
• Request Review Copy
Volume: 1202014; 69 pp
MSC: 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 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.

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 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
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