Ergodic theory and fractal geometry
Author(s)
Bibliographic Information
Ergodic theory and fractal geometry
(Regional conference series in mathematics, no. 120)
Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, c2014
Available at 33 libraries
  Aomori
  Iwate
  Miyagi
  Akita
  Yamagata
  Fukushima
  Ibaraki
  Tochigi
  Gunma
  Saitama
  Chiba
  Tokyo
  Kanagawa
  Niigata
  Toyama
  Ishikawa
  Fukui
  Yamanashi
  Nagano
  Gifu
  Shizuoka
  Aichi
  Mie
  Shiga
  Kyoto
  Osaka
  Hyogo
  Nara
  Wakayama
  Tottori
  Shimane
  Okayama
  Hiroshima
  Yamaguchi
  Tokushima
  Kagawa
  Ehime
  Kochi
  Fukuoka
  Saga
  Nagasaki
  Kumamoto
  Oita
  Miyazaki
  Kagoshima
  Okinawa
  Korea
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  United Kingdom
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  Switzerland
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  United States of America
Note
"With support from the National Science Foundation"
"NSF/CBMS Regional Conference in the Mathematical Sciences on Ergodic Methods in the Theory of Fractals, held at Kent State University, June 18-23, 2011"--T.p. verso
Includes bibliographical references (p. 67) and index
Description and Table of Contents
Description
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.
Table of Contents
Introduction to fractals
Dimension Trees and fractals
Invariant sets
Probability trees
Galleries
Probability trees revisited
Elements of ergodic theory
Galleries of trees
General remarks on Markov systems
Markov operator $\mathcal{T}$ and measure preserving transformation $T$
Probability trees and galleries
Ergodic theorem and the proof of the main theorem
An application: The $k$-lane property
Dimension and energy Dimension conservation
Ergodic theorem for sequences of functions
Dimension conservation for homogeneous fractals: The main steps in the proof
Verifying the conditions of the ergodic theorem for sequences of functions
Bibliography
Index
by "Nielsen BookData"