Ergodic theory and fractal geometry
著者
書誌事項
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
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注記
"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
内容説明・目次
内容説明
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.
目次
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
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