Recurrent point

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

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. 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), "$\bar P=\bar R$ for maps of the interval", Proceedings of the American Mathematical Society 79 (2): 316–318, doi:10.2307/2043258, MR 565362.
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).

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