topologically transitive

A continuous surjection $f$ on a topological space $X$ to itself is topologically transitive if for every pair of open sets $U$ and $V$ in $X$ there is an integer $n>0$ such that $f^n(U)\cap V\ne \mathrm{\varnothing }$, where $f^n$ denotes the $n$-th iterate of $f$.

If for every pair of open sets $U$ and $V$ there is an integer $N$ such that $f^n(U)\cap V\ne \mathrm{\varnothing }$ for each $n>N$, we say that $f$ is topologically mixing.

If $X$ is a compact metric space, then $f$ is topologically transitive if and only if there exists a point $x\in X$ with a dense orbit, i.e. such that $𝒪(x,f)=\{f^n(x):n\in \mathbb{N}\}$ is dense in $X$.

Title	topologically transitive
Canonical name	TopologicallyTransitive
Date of creation	2013-03-22 13:41:05
Last modified on	2013-03-22 13:41:05
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	5
Author	Koro (127)
Entry type	Definition
Classification	msc 37B99
Classification	msc 54H20
Defines	topologically mixing
Defines	topological mixing