Ergodic theory and fractal geometry

Author(s)

• Furstenberg, Hillel
• Conference Board of the Mathematical Sciences

Bibliographic Information

Ergodic theory and fractal geometry

Hillel Furstenberg

（Regional conference series in mathematics, no. 120）

Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, c2014

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"

Details

• NCID: BB16388082
• ISBN:
  • 9781470410346
• LCCN: 2014010556
• Country Code: us
• Title Language Code: eng
• Text Language Code: eng
• Place of Publication: Providence, R.I.
• Pages/Volumes: ix, 69 p.
• Size: 26 cm
• Classification:
  • LCC : QA313
  • DC23 : 515/.48
• Subject Headings:
  • LCSH : Ergodic theory -- Congresses
  • LCSH : Fractals -- Congresses
  • FREE : Measure and integration -- Classical measure theory -- Fractals. msc
  • FREE : Dynamical systems and ergodic theory -- Ergodic theory -- Ergodic theorems, spectral theory, Markov operators. msc
  • FREE : Functions of a complex variable -- Entire and meromorphic functions, and related topics -- Functional equations in the complex domain, iteration and composition of analytic functions. msc
  • FREE : Dynamical systems and ergodic theory -- Complex dynamical systems -- Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations. msc
  • FREE : Operator theory -- General theory of linear operators -- Ergodic theory. msc
• Parent Bibliography ID:
  • BA0000944X