# Recurrence in Topological Dynamics

## Furstenberg Families and Ellis Actions

• EthanĀ Akin
Book

Part of the The University Series in Mathematics book series (USMA)

1. Front Matter
Pages i-ix
2. Ethan Akin
Pages 1-10
3. Ethan Akin
Pages 11-22
4. Ethan Akin
Pages 23-51
5. Ethan Akin
Pages 53-74
6. Ethan Akin
Pages 75-100
7. Ethan Akin
Pages 101-131
8. Ethan Akin
Pages 133-153
9. Ethan Akin
Pages 155-192
10. Ethan Akin
Pages 193-228
11. Back Matter
Pages 229-265

### Introduction

In the long run of a dynamical system, after transient phenomena have passed away, what remains is recurrence. An orbit is recurrent when it returns repeatedly to each neighborhood of its initial position. We can sharpen the concept by insisting that the returns occur with at least some prescribed frequency. For example, an orbit lies in some minimal subset if and only if it returns almost periodically to each neighborhood of the initial point. That is, each return time set is a so-called syndetic subset ofT= the positive reals (continuous time system) or T = the positive integers (discrete time system). This is a prototype for many of the results in this book. In particular, frequency is measured by membership in a family of subsets of the space modeling time, in this case the family of syndetic subsets of T. In applying dynamics to combinatorial number theory, Furstenberg introduced a large number of such families. Our first task is to describe explicitly the calculus of families implicit in Furstenberg's original work and in the results which have proliferated since. There are general constructions on families, e. g. , the dual of a family and the product of families. Other natural constructions arise from a topology or group action on the underlying set. The foundations are laid, in perhaps tedious detail, in Chapter 2. The family machinery is then applied in Chapters 3 and 4 to describe family versions of recurrence, topological transitivity, distality and rigidity.

### Keywords

Compactification DEX Volume dynamical systems ergodic theory ergodicity mixing number theory semigroup set tool

#### Authors and affiliations

• EthanĀ Akin
• 1
1. 1.The City CollegeNew YorkUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4757-2668-8
• Copyright Information Springer-Verlag US 1997
• Publisher Name Springer, Boston, MA
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• Print ISBN 978-1-4419-3272-3
• Online ISBN 978-1-4757-2668-8
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