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.
Rezensionen / Stimmen
Fractals are beautiful and complex geometric objects. Their study, pioneered by Benoit 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 Szemeredi, Renyi Institute of Mathematics, Budapest
Reihe
Sprache
Verlagsort
Zielgruppe
Maße
Höhe: 254 mm
Breite: 178 mm
Gewicht
ISBN-13
978-1-4704-1034-6 (9781470410346)
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Schweitzer Klassifikation
Hillel Furstenberg, The Hebrew University of Jerusalem, Israel
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
