Recurrent point

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In mathematics, a recurrent point for a function f is a point that is in its own limit set by f. Any neighborhood containing the recurrent point will also contain (a countable number of) iterates of it as well.

Definition[edit]

Let be a Hausdorff space and a function. A point is said to be recurrent (for ) if , i.e. if belongs to its -limit set. This means that for each neighborhood of there exists such that .[1]

The set of recurrent points of is often denoted and is called the recurrent set of . Its closure is called the Birkhoff center of ,[2] and appears in the work of George David Birkhoff on dynamical systems.[3][4]

Every recurrent point is a nonwandering point,[1] hence if is a homeomorphism and is compact, then is an invariant subset of the non-wandering set of (and may be a proper subset).

References[edit]

  1. ^ a b Irwin, M. C. (2001), Smooth dynamical systems, Advanced Series in Nonlinear Dynamics, 17, World Scientific Publishing Co., Inc., River Edge, NJ, p. 47, doi:10.1142/9789812810120, ISBN 981-02-4599-8, MR 1867353 .
  2. ^ Hart, Klaas Pieter; Nagata, Jun-iti; Vaughan, Jerry E. (2004), Encyclopedia of general topology, Elsevier, p. 390, ISBN 0-444-50355-2, MR 2049453 .
  3. ^ Coven, Ethan M.; Hedlund, G. A. (1980), " for maps of the interval", Proceedings of the American Mathematical Society, 79 (2): 316–318, doi:10.2307/2043258, MR 565362 .
  4. ^ Birkhoff, G. D. (1927), "Chapter 7", Dynamical Systems, Amer. Math. Soc. Colloq. Publ., 9, Providence, R. I.: American Mathematical Society . As cited by Coven & Hedlund (1980).


This article incorporates material from Recurrent point on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.