Recurrent point

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

Definition

Let $X$ be a Hausdorff space and $f\colon X\to X$ a function. A point $x\in X$ is said to be recurrent (for $f$ ) if $x\in \omega(x)$ , i.e. if $x$ belongs to its $\omega$ -limit set. This means that for each neighborhood $U$ of $x$ there exists $n>0$ such that $f^n(x)\in U$ .^[1]

The set of recurrent points of $f$ is often denoted $R(f)$ and is called the recurrent set of $f$ . Its closure is called the Birkhoff center of $f$ ,^[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 $f$ is a homeomorphism and $X$ is compact, then $R(f)$ is an invariant subset of the non-wandering set of $f$ (and may be a proper subset).

References

↑ 1.0 1.1 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 .
↑ Hart, Klaas Pieter; Nagata, Jun-iti; Vaughan, Jerry E. (2004), Encyclopedia of general topology, Elsevier, p. 390, ISBN 0-444-50355-2, MR 2049453 .
↑ Coven, Ethan M.; Hedlund, G. A. (1980), " $\bar P=\bar R$ for maps of the interval", Proceedings of the American Mathematical Society 79 (2): 316–318, doi:10.2307/2043258, MR 565362 .
↑ 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).

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