This book presents the expanded notes from ten lectures given by the author at the NSF/CBMS conference held at California State University (Bakersfield). The author describes what he calls single orbit dynamics, which is an approach to the analysis of dynamical systems via the study of single orbits, rather than the study of a system as a whole. He presents single orbit interpretations of several areas of topological dynamics and ergodic theory and some new applications of dynamics to graph theory.

In the concluding lectures, single orbit approaches to generalizations of the Shannon-Breiman-McMillan theorem and related problems of compression and universal coding are presented. Complete proofs and illuminating discussions are included and references for further study are given. Some of the material appears here for the first time in print.

Readership

Graduate students and research mathematicians interested in dynamical systems.

Reviews

"These notes give a unique tour through modern dynamical systems in the broadest sense. The perspective is unique, and the exposition is clear enough for the notes to be of value to graduate students in ergodic theory and dynamical systems or probability, as well as being essential reading for experts."

-- Mathematical Reviews

Table of Contents

• What is single orbit dynamics
• Topological dynamics
• Invariant measures, ergodicity and unique ergodicity
• Ergodic and uniquely ergodic orbits
• Translation invariant graphs and recurrence
• Patterns in large sets
• Entropy and disjointness
• What is randomness?
• Recurrence rates and entropy
• Universal schemes