Topological transitivity

In mathematics , one speaks of the topological transitivity of a mapping when it "swirls up" a metric space . In the literature, topological transitivity is therefore often referred to as mixing:

"If U is any open set in the domain of the function, then some point of U will eventually land in every neighborhood of every point in the domain under iteration of the function."

- Holmgren

Topological transitivity is particularly with regard to the diagnosis of chaos in the sense of Devaney important: A picture is chaotic if it is topologically transitive and the amount of the periodic points of tightly in lies. ${\displaystyle f\colon X\to X}$ ${\displaystyle f}$ ${\displaystyle X}$

definition

Let it be a metric space and ${\displaystyle X}$

${\displaystyle f\colon X\to X}$

a continuous mapping of this space into itself. Then is topologically transitive if for any two non-empty open subsets of true ${\displaystyle f}$ ${\displaystyle U,V}$ ${\displaystyle X}$

${\displaystyle \exists \ n\in \mathbb {N} :f^{n}(U)\cap V\neq \emptyset }$

in which

${\displaystyle f^{n}(U)=\left\{f^{n}(x)|x\in U\right\}={\big \{}\underbrace {f\circ \cdots \circ f} _{n}(x)|x\in U{\big \}}}$

discussion

As indicated above, topological transitivity and density of the periodic points are the two properties that have to be demanded when speaking of chaos in the Devaney sense. Devaney also called for sensitive dependence on the initial conditions. However, Banks et al. prove that this property already follows from the other two.

The proof of topological transitivity is i. A. laborious, since it has to be shown for arbitrary open sets that they are mixed. In this context it is helpful to say that the existence of a point in its orbit is sufficient ${\displaystyle U,V}$ ${\displaystyle x}$ ${\displaystyle X}$

${\displaystyle {\mathcal {O}}_{f}(x)=\left\{f^{n}(x)|n\in \mathbb {N} \right\}}$

is dense in so that it is topologically transitive. ${\displaystyle X}$ ${\displaystyle f}$

example

We look at the picture

${\displaystyle f\colon S^{1}\to S^{1},\ \theta \mapsto 2\theta }$

on the unit circle . Then: is topologically transitive. Because it applies: ${\displaystyle S^{1}=\mathbb {R} /2\pi \mathbb {Z} }$ ${\displaystyle f}$

${\displaystyle f^{n}(\theta )=2^{n}\theta ({\hbox{ mod }}2\pi )\ \forall \ n\in \mathbb {N} ,\ \theta \in [0,2\pi )}$

From this we can see that the mapping is expansive and that every piece of arc , no matter how small, expands so strongly that it finally covers the entire unit circle and thus every other open interval as well. ${\displaystyle f}$ ${\displaystyle n}$

literature

1. ^ RA Holmgren: A First Course in Discrete Dynamical Systems , Springer Verlag, New York 2006, ISBN 0387947809
2. ↑ Banks et al .: Chaos. A mathematical introduction , Cambridge University Press, Cambridge 2003, ISBN 0521531047