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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
Ergodic Theory and Fractal Geometry
Softcover ISBN:  978-1-4704-1034-6
Product Code:  CBMS/120
List Price: $36.00
MAA Member Price: $32.40
AMS Member Price: $28.80
eBook ISBN:  978-1-4704-1854-0
Product Code:  CBMS/120.E
List Price: $34.00
MAA Member Price: $30.60
AMS Member Price: $27.20
Softcover ISBN:  978-1-4704-1034-6
eBook: ISBN:  978-1-4704-1854-0
Product Code:  CBMS/120.B
List Price: $70.00 $53.00
MAA Member Price: $63.00 $47.70
AMS Member Price: $56.00 $42.40
Ergodic Theory and Fractal Geometry
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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
Softcover ISBN:  978-1-4704-1034-6
Product Code:  CBMS/120
List Price: $36.00
MAA Member Price: $32.40
AMS Member Price: $28.80
eBook ISBN:  978-1-4704-1854-0
Product Code:  CBMS/120.E
List Price: $34.00
MAA Member Price: $30.60
AMS Member Price: $27.20
Softcover ISBN:  978-1-4704-1034-6
eBook ISBN:  978-1-4704-1854-0
Product Code:  CBMS/120.B
List Price: $70.00 $53.00
MAA Member Price: $63.00 $47.70
AMS Member Price: $56.00 $42.40
  • 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.

    Readership

    Graduate 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
  • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Accessibility – to request an alternate format of an AMS title
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.

Readership

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
Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
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