Wandering point

In the theory of dynamic systems , a non-moving point (also: non- moving point ) is a point whose orbit returns as close as desired to the starting position and the orbites of an entire area around the point come as close as desired to this point. (In particular, periodic points are non- wandering .) Similarly, a wandering point is a point for which an entire environment never returns to that environment. The set of migrating or non-migrating points is referred to as the migrating set or non-migrating set .

Definition of discrete dynamic systems

It is a metric space and a constant transformation . ${\displaystyle X}$ ${\displaystyle f\colon X\to X}$

A point is a wandering point if there is a neighborhood such that ${\displaystyle x\in X}$ ${\displaystyle x\in U}$

${\displaystyle f^{n}(U)\cap U=\emptyset }$

for everyone . ${\displaystyle n>0}$

A point is a non-migrating point if there is one with for every environment ${\displaystyle x\in X}$ ${\displaystyle x\in U}$ ${\displaystyle n>0}$

${\displaystyle f^{n}(U)\cap U\not =\emptyset }$

gives.

Definition of rivers

It is a manifold and a river . ${\displaystyle M}$ ${\displaystyle \phi \colon M\times \mathbb {R} \to M}$

A point is a migrating point if there is a neighborhood of and such that ${\displaystyle x\in M}$ ${\displaystyle U}$ ${\displaystyle x}$ ${\displaystyle T>0}$

${\displaystyle \phi _{t}(U)\cap U=\emptyset }$

for everyone . ${\displaystyle t\geq T}$

A point is a non-migratory point when for every environment and for every one with ${\displaystyle x\in M}$ ${\displaystyle x\in U}$ ${\displaystyle T>0}$ ${\displaystyle t\geq T}$

${\displaystyle \phi _{t}(U)\cap U\not =\emptyset }$

gives.

properties

The set of the non-migrating point is closed , invariant and contains all - Limes sets . It contains all recurrent points , but not every non-migrating point has to be recurrent. ${\displaystyle \Omega (f)}$ ${\displaystyle \omega }$

If is compact , then is . ${\displaystyle M}$ ${\displaystyle \Omega (f)\not =\emptyset }$

literature

• Manfred Denker : Introduction to the Analysis of Dynamic Systems. Springer textbook. Springer-Verlag, Berlin 2005, ISBN 3-540-20713-9 .

Web links

• Nonwandering (MathWorld)