Chaos on the Interval
About this Title
Sylvie Ruette, Université Paris-Sud, Orsay, France
Publication: University Lecture Series
Publication Year: 2017; Volume 67
ISBNs: 978-1-4704-2956-0 (print); 978-1-4704-3759-6 (online)
MathSciNet review: MR3616574
MSC: Primary 37-02; Secondary 37D45, 37E05
The aim of this book is to survey the relations between the various kinds of chaos and related notions for continuous interval maps from a topological point of view. The papers on this topic are numerous and widely scattered in the literature; some of them are little known, difficult to find, or originally published in Russian, Ukrainian, or Chinese. Dynamical systems given by the iteration of a continuous map on an interval have been broadly studied because they are simple but nevertheless exhibit complex behaviors. They also allow numerical simulations, which enabled the discovery of some chaotic phenomena. Moreover, the “most interesting” part of some higher-dimensional systems can be of lower dimension, which allows, in some cases, boiling it down to systems in dimension one.
Some of the more recent developments such as distributional chaos, the relation between entropy and Li-Yorke chaos, sequence entropy, and maps with infinitely many branches are presented in book form for the first time. The author gives complete proofs and addresses both graduate students and researchers.
Graduate students and researchers interested in one-dimensional dynamical systems.
Table of Contents
- Notation and basic tools
- Links between transitivity, mixing and sensitivity
- Periodic points
- Topological entropy
- Chaos in the sense of Li-Yorke, scrambled sets
- Other notions related to Li-Yorke pairs: Generic and dense chaos, distributional chaos
- Chaotic subsystems
- Appendix: Some background in topology