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 .
Discrete dynamic systems
It is a discrete dynamic system. A point is called recurrent if there are infinitely many with each
gives.
Equivalent: there is a partial sequence with
- .
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
gives.
Equivalent: there is an infinite sequence with
- .
Group effects
It is a group effect. A point is recurrent, if for every a sequence pairs of different elements with
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 .
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
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.
- If the orbit of densely located, then recurrent.
Birkhoff's recurring theorem
Every continuous dynamic system in a compact space has almost periodic and consequently recurrent points.
Poincaré recurring clause
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. .
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.
Examples
- Be and a turn , then each point is recurrent.
- Let be a topological group and a co-compact lattice . Be an element from the center of . The image
- defines a dynamic system and from Birkhoff's recurrence theorem it follows that every point is recurrent.
- Applying the previous example with and results in Kronecker's approximation theorem .
properties
- Any recurrent point is non-migrating .
- The set of recurrent points is invariant below . Your conclusion is the Birkhoff crowd (English: Birkhoff center).
- 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
- ^ Feres, Katok: Ergodic theory and dynamics of G-actions , page 19
- ^ Theorem 3.4.1 in: Katok, Hasselblatt: Principal structures. Handbook of dynamical systems, Vol. 1A, 1-203, North-Holland, Amsterdam, 2002.