Recurrent point (original) (raw)

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Mathematical concept

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.

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

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

  1. ^ a b 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), " P ¯ = R ¯ {\displaystyle {\bar {P}}={\bar {R}}} {\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).

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