Topological mixing

Remarks

Every topologically mixing map is topologically transitive. Example: the doubling map $f(x) = 2x \bmod 1$ on the circle $\R / \Z$. A nonempty open set $U$ contains an open arc of length $\ell > 0$; choose $k$ with $2^k \ell > 1$. The map $f^n$ multiplies lengths of small arcs by $2^n$, so for $n \ge k$ the image $f^n(U)$ is the whole circle and meets every $V$. Non-example: the rotation $f(x) = x + \alpha \bmod 1$ with $\alpha$ irrational, and $U = V = (-1/8, 1/8)$. Here $f^n(U) \cap U \neq \emptyset$ only when $n\alpha$ is within $1/4$ of an integer. By Weyl's equidistribution theorem, about half of all $n$ fail this, so there is no $N$ as required.