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 $\mathcal{O}(x,f)=\{f^n(x): n\in \N\}$ is dense in $X$