Description
Book SynopsisTrade ReviewFractals are beautiful and complex geometric objects. Their study, pioneered by Benoît 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 Szemerédi, Rényi Institute of Mathematics, Budapest
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
