Recurrent point

The terms recurrent points and recurrent orbits are used in the mathematical theory of ( dimensionally conserving or even continuous ) dynamic systems . The recurrence of a point under a river (or more generally a group effect ) means that this point returns infinitely often to the vicinity of its starting position.

definition

We first give the definition for discrete dynamic systems , then the very similar definitions for continuous dynamic systems (flows) and for general group effects.

Notations: A group action of a group on a metric space is given by a map , wherein one of the image with designated. Discrete dynamic systems correspond to the special case and flows correspond to the special case . In the case , we denote the mapping with and its -th iteration for , i.e. the mapping . In the case of continuous dynamic systems (flows) we denote for and . ${\displaystyle G}$ ${\displaystyle (X,d)}$ ${\displaystyle \Phi \colon G\times X\to X}$ ${\displaystyle (g,x)}$ ${\displaystyle gx}$ ${\displaystyle G=\mathbb {Z} }$ ${\displaystyle G=\mathbb {R} }$ ${\displaystyle G=\mathbb {Z} }$ ${\displaystyle T\colon X\to X}$ ${\displaystyle x\mapsto \Phi (1,x)}$ ${\displaystyle T^{n}\colon X\to X}$ ${\displaystyle n}$ ${\displaystyle n\in N}$ ${\displaystyle x\mapsto \Phi (n,x)}$ ${\displaystyle \phi _{t}(x):=\Phi (t,x)}$ ${\displaystyle t\in \mathbb {R} }$ ${\displaystyle x\in X}$

Discrete dynamic systems

It is a discrete dynamic system. A point is called recurrent if there are infinitely many with each ${\displaystyle (X,T)}$ ${\displaystyle x\in X}$ ${\displaystyle \epsilon >0}$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle d(x,T^{n}x)<\epsilon }$

gives.

Equivalent: there is a partial sequence with ${\displaystyle (n_{j})_{j}\subset \mathbb {N} }$

${\displaystyle \lim _{j\to \infty }T^{n_{j}}x=x}$.

The orbit of a recurrent point is called the recurrent orbit.

Continuous dynamic systems

It is a river. A point is recurrent, if for every one going towards infinity result with ${\displaystyle (\phi _{t}\colon X\to X)_{t\in \mathbb {R} }}$ ${\displaystyle x\in X}$ ${\displaystyle \epsilon >0}$ ${\displaystyle t_{i}\to \infty }$

${\displaystyle d(x,\phi _{t_{i}}x)<\epsilon }$

gives.

Equivalent: there is an infinite sequence with ${\displaystyle t_{i}\to \infty }$

${\displaystyle \lim _{i\to \infty }\phi _{t_{i}}x=x}$.

Group effects

It is a group effect. A point is recurrent, if for every a sequence pairs of different elements with ${\displaystyle \Phi \colon G\times X\to X}$ ${\displaystyle x\in X}$ ${\displaystyle \epsilon >0}$ ${\displaystyle g_{n}}$ ${\displaystyle G}$

${\displaystyle \lim _{n\to \infty }d(x,g_{n}x)=0}$

gives. The group effect is called recurrent if the recurrent points are close together.

Dimensionally preserving dynamic systems

For dimensionally conserving dynamic systems , the recurrence condition can also be formulated as follows. Let it be a dimensional space and a dimensionally preserving illustration . The mapping is called recurrent if there is an infinite number with for every set with and for almost all . ${\displaystyle (X,\Sigma ,\mu )}$ ${\displaystyle f\colon X\to X}$ ${\displaystyle A\subset X}$ ${\displaystyle \mu (A)>0}$ ${\displaystyle \mu }$ ${\displaystyle x\in A}$ ${\displaystyle n\in \mathbb {Z} }$ ${\displaystyle f^{n}(x)\in A}$

Similarly, one can define recurrence for measure-maintaining effects of any group. The effect of a group is called recurrent if for every crowd with and for almost all the crowd ${\displaystyle G}$ ${\displaystyle A\subset X}$ ${\displaystyle \mu (A)>0}$ ${\displaystyle \mu }$ ${\displaystyle x\in A}$

${\displaystyle g\in G\colon gx\in A}$

is not relatively compact .

Special cases

Are special cases of recurrent points

• Fixed points
• Periodic points
• Almost periodic points, i.e. H. so that for all the set is a syndetic set , that is, has bounded gaps. ${\displaystyle x\in X}$ ${\displaystyle \epsilon >0}$ ${\displaystyle \left\{n\in \mathbb {N} \colon d(x,T^{n}x)<\epsilon \right\}}$
• If the orbit of densely located, then recurrent. ${\displaystyle x}$ ${\displaystyle x}$

Birkhoff's recurring theorem

Every continuous dynamic system in a compact space has almost periodic and consequently recurrent points.

Poincaré recurring clause

→ Main article : Recurrence theorem

Poincaré's recurrence theorem says: If has finite volume, then every measure-preserving map has recurrent points. Furthermore, the set of recurrent points has full measure, i.e. H. . ${\displaystyle X}$ ${\displaystyle f\colon X\to X}$ ${\displaystyle \mu (R(f))=\mu (X)}$

This theorem has a more general version for mass-maintaining group effects. Is a non-compact, locally compact group , which the second countable fulfilled and on a measure space with am working. Then the effect is recurrent. ${\displaystyle G}$ ${\displaystyle (X,\sigma ,\mu )}$ ${\displaystyle \mu (X)<\infty }$

Examples

• Be and a turn , then each point is recurrent. ${\displaystyle X=S^{1}}$ ${\displaystyle f\colon X\to X}$
• Let be a topological group and a co-compact lattice . Be an element from the center of . The image ${\displaystyle G}$ ${\displaystyle \Gamma \subset G}$ ${\displaystyle g\in Z(G)\subset G}$ ${\displaystyle G}$

${\displaystyle x\to gx}$

defines a dynamic system and from Birkhoff's recurrence theorem it follows that every point is recurrent. ${\displaystyle G/\Gamma }$

• Applying the previous example with and results in Kronecker's approximation theorem . ${\displaystyle G=\mathbb {R} ^{d}}$ ${\displaystyle \Gamma =\mathbb {Z} ^{d}}$

properties

• Any recurrent point is non-migrating .
• The set of recurrent points is invariant below . Your conclusion is the Birkhoff crowd (English: Birkhoff center). ${\displaystyle R(f)}$ ${\displaystyle f}$
• Poisson stability : The property of a point to be recurrent is stable under slight changes in the dynamic system.

Web links

• Terence Tao: Minimal dynamical systems, recurrence, and the Stone-Čech compactification
• Recurrent point (Encyclopedia of Mathematics)

Individual evidence

1. ^ Feres, Katok: Ergodic theory and dynamics of G-actions , page 19
2. ^ Theorem 3.4.1 in: Katok, Hasselblatt: Principal structures. Handbook of dynamical systems, Vol. 1A, 1-203, North-Holland, Amsterdam, 2002.