Ergodic theory and fractal geometry / Hillel Furstenberg.

"Support from the National Science Foundation."

Includes bibliographical references (page 67) and index.

1. Introduction to fractals
2. Dimension
3. Trees and fractals
4. Invariant sets
5. Probability trees
6. Galleries
7. Probability trees revisited
8. Elements of ergodic theory
9. Galleries of trees
10. General remarks on Markov systems
11. Markov operator mathcal{T} and measure preserving transformation {T}
12. Probability trees and galleries
13. Ergodic theorem and the proof of the main theorem
14. An application: The $k$-lane property
15. Dimension and energy
16. Dimension conservation
17. Ergodic theorem for sequences of functions
18. Dimension conservation for homogeneous fractals: The main steps in the proof
19. Verifying the conditions of the ergodic theorem for sequences of functions
Bibliography
Index.

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.