Topological transitivity

Remarks

An equivalent form is $U \cap f^{-n}(V) \neq \emptyset$, since both say that some point of $U$ is sent into $V$ by $f^n$. Let $X$ be a compact metric space without isolated points. Then $f$ is topologically transitive if and only if some forward orbit $\set{f^n(x) : n \ge 0}$ is dense in $X$. In that case the points with dense forward orbit form a dense $G_\delta$ set. Example: let $f(x) = x + \alpha \bmod 1$ on the circle $\R / \Z$ with $\alpha$ irrational. Every forward orbit is dense, so for nonempty open $U, V$ and a point $x \in U$, some $f^n(x)$ with $n \ge 1$ lies in $V$. Non-example: the identity map of $[0, 1]$, with $U = (0, 1/3)$ and $V = (2/3, 1)$, has $f^n(U) \cap V = U \cap V = \emptyset$ for all $n$.