Ergodic theory and fractal geometry
 Responsibility
 Hillel Furstenberg.
 Publication
 Providence, Rhode Island : Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, [2014]
 Copyright notice
 ©2014
 Physical description
 ix, 69 pages : illustrations ; 26 cm.
 Series
 Regional conference series in mathematics no. 120.
At the library
Science Library (Li and Ma)
Stacks
Call number  Status 

QA1 .R33 NO.120  Unknown 
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Description
Creators/Contributors
 Author/Creator
 Furstenberg, Harry, author.
 Contributor
 Conference Board of the Mathematical Sciences.
 National Science Foundation (U.S.)
Contents/Summary
 Bibliography
 Includes bibliographical references (page 67) and index.
 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.
 (source: Nielsen Book Data)
 Summary

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 dynamicsergodic theoryto 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.
(source: Nielsen Book Data)
Subjects
 Subject
 Ergodic theory > Congresses.
 Fractals > Congresses.
 Measure and integration  Classical measure theory  Fractals.
 Dynamical systems and ergodic theory  Ergodic theory  Ergodic theorems, spectral theory, Markov operators.
 Functions of a complex variable  Entire and meromorphic functions, and related topics  Functional equations in the complex domain, iteration and composition of analytic functions.
 Dynamical systems and ergodic theory  Complex dynamical systems  Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations.
 Operator theory  General theory of linear operators  Ergodic theory.
 Ergodic theory.
 Fractals.
Bibliographic information
 Publication date
 2014
 Copyright date
 2014
 Series
 Regional Conference series in mathematics / Conference Board of the Mathematical Sciences ; number 120
 Note
 "Notes based on a series of lectures delivered at Kent State University in 2011"Preface.
 "NSF/CBMS Regional Conference in the Mathematical Sciences on Ergodic Methods in the Theory of Fractals, held at Kent State University, June 1823, 2011"Verso of title page.
 "With support from the National Science Foundation."
 ISBN
 9781470410346 (softcover : alk. paper)
 1470410346 (softcover : alk. paper)