Recurrent point

Last updated May 04, 2019

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.

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they can be used to understand the long term behavior of a dynamical system.

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]

In topology and related branches of mathematics, a Hausdorff space, separated space or T₂ space is a topological space where for any two distinct points there exists a neighbourhood of each which is disjoint from the neighbourhood of the other. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T₂) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters.

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]

George David Birkhoff was an American mathematician best known for what is now called the ergodic theorem. Birkhoff was one of the most important leaders in American mathematics in his generation, and during his time he was considered by many to be the preeminent American mathematician.

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.

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).

Homeomorphism In mathematics, isomorphism of topological spaces

In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word homeomorphism comes from the Greek words ὅμοιος (homoios) = similar or same and μορφή (morphē) = shape, form, introduced to mathematics by Henri Poincaré in 1895.

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Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism.

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In mathematical analysis, a function of bounded variation, also known as $BV$ function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single variable, being of bounded variation means that the distance along the direction of the $y$ -axis, neglecting the contribution of motion along $x$ -axis, traveled by a point moving along the graph has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function, but can be every intersection of the graph itself with a hyperplane parallel to a fixed $x$ -axis and to the $y$ -axis.

In mathematics, the oscillation of a function or a sequence is a number that quantifies how much a sequence or function varies between its extreme values as it approaches infinity or a point. As is the case with limits there are several definitions that put the intuitive concept into a form suitable for a mathematical treatment: oscillation of a sequence of real numbers, oscillation of a real valued function at a point, and oscillation of a function on an interval.

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References

1 2 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 0565362 .
↑ 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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[irwin-1] 1 2 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 0565362 .

[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).