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 $f^n(x)\in U$ .{{citation

| last = Irwin | first = M. C.

| doi = 10.1142/9789812810120

| isbn = 981-02-4599-8

| mr = 1867353

| page = 47

| publisher = World Scientific Publishing Co., Inc., River Edge, NJ

| series = Advanced Series in Nonlinear Dynamics

| title = Smooth dynamical systems

| url = https://books.google.com/books?id=bu9k9-NonpoC&pg=PA47

| volume = 17

| year = 2001}}.

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 $f$ ,{{citation

| last1 = Hart | first1 = Klaas Pieter

| last2 = Nagata | first2 = Jun-iti

| last3 = Vaughan | first3 = Jerry E.

| isbn = 0-444-50355-2

| mr = 2049453

| page = 390

| publisher = Elsevier

| title = Encyclopedia of general topology

| url = https://books.google.com/books?id=JWyoCRkLFAkC&pg=PA390

| year = 2004}}. and appears in the work of George David Birkhoff on dynamical systems.{{citation

| last1 = Coven | first1 = Ethan M.

| last2 = Hedlund | first2 = G. A. | author2-link = Gustav A. Hedlund

| doi = 10.1090/S0002-9939-1980-0565362-0 | doi-access=free

| jstor=2043258

| issue = 2

| journal = Proceedings of the American Mathematical Society

| mr = 565362

| pages = 316–318

| title = $\bar P=\bar R$ for maps of the interval

| volume = 79

| year = 1980}}.{{citation|first=G. D.|last=Birkhoff|authorlink=George David Birkhoff|title=Dynamical Systems|volume=9|series=Amer. Math. Soc. Colloq. Publ.|publisher=American Mathematical Society|location=Providence, R. I.|contribution=Chapter 7|year=1927}}. As cited by {{harvtxt|Coven|Hedlund|1980}}.

Every recurrent point is a nonwandering point, 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

Category:Limit sets