
Recurrence in Topological Dynamics
Furstenberg Families and Ellis Actions
Ethan Akin(Author)
Springer (Publisher)
Published on 6. December 2010
Book
Paperback/Softback
X, 266 pages
978-1-4419-3272-3 (ISBN)
Description
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.
More details
Series
Edition
1st ed. Softcover of orig. ed. 1997
Language
English
Place of publication
New York
United States
Target group
Professional and scholarly
Research
Illustrations
X, 266 p.
Dimensions
Height: 235 mm
Width: 155 mm
Thickness: 16 mm
Weight
423 gr
ISBN-13
978-1-4419-3272-3 (9781441932723)
DOI
10.1007/978-1-4757-2668-8
Schweitzer Classification
Other editions
Additional editions

Book
07/1997
Plenum Publishing Co.,N.Y.
€106.99
Shipment within 10-15 days
Content
1. Monoid Actions.- 2. Furstenberg Families.- 3. Recurrence.- 4. Transitive and Central Systems.- 5. Compactifications.- 6. Ellis Semigroups and Ellis Actions.- 7. Semigroups and Families.- 8. Equicontinuity.- Appendix. Semicontinuous Relations and Almost Open Maps.- References.