In mathematics, a recurrent point for a function ''f'' is a point that is in its own

limit set 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 c ...

by ''f''. Any neighborhood containing the recurrent point will also contain (a

countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...

number of) iterates of it as well.

Definition

Let

X

be a

Hausdorff space 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 exist neighbourhoods of each which are disjoint from each other. Of the many ...

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

. This means that for each neighborhood

U

x

there exists

n>0

such that

f^n(x)\in U

.. 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

, and appears in the work of

George David Birkhoff George David Birkhoff (March 21, 1884 – November 12, 1944) 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 duri ...

dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...

s.. As cited by . Every recurrent point is a

nonwandering point In dynamical systems and ergodic theory, the concept of a wandering set formalizes a certain idea of movement and mixing. When a dynamical system has a wandering set of non-zero measure, then the system is a dissipative system. This is the opposit ...

, hence if

f

is a

homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...

and

X

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

, then

R(f)

is an invariant subset of the non-wandering set of

f

(and may be a

proper subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...

References

Limit sets {{topology-stub