# 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 $f^{n}(U)\cap V\neq\emptyset$, 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 $f^{n}(U)\cap V\neq\emptyset$ 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 $\mathcal{O}(x,f)=\{f^{n}(x):n\in\mathbb{N}\}$ is dense in $X$.

Title topologically transitive TopologicallyTransitive 2013-03-22 13:41:05 2013-03-22 13:41:05 Koro (127) Koro (127) 5 Koro (127) Definition msc 37B99 msc 54H20 topologically mixing topological mixing