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

The set of recurrent points of ${\displaystyle f}$ is often denoted ${\displaystyle R(f)}$ and is called the recurrent set of ${\displaystyle f}$. Its closure is called the Birkhoff center of ${\displaystyle f}$, and appears in the work of George David Birkhoff on dynamical systems.

Every recurrent point is a nonwandering point, hence if ${\displaystyle f}$ is a homeomorphism and ${\displaystyle X}$ is compact, then ${\displaystyle R(f)}$ is an invariant subset of the non-wandering set of ${\displaystyle f}$ (and may be a proper subset).

References

1. ^ Irwin, M. C. (2001), Smooth dynamical systems, Advanced Series in Nonlinear Dynamics, vol. 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), "${\displaystyle {\bar {P}}={\bar {R}}}$ for maps of the interval", Proceedings of the American Mathematical Society, 79 (2): 316–318, doi:10.1090/S0002-9939-1980-0565362-0, JSTOR 2043258, MR 0565362.
4. Birkhoff, G. D. (1927), "Chapter 7", Dynamical Systems, Amer. Math. Soc. Colloq. Publ., vol. 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.

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