Point transitivity ($\overline{\{f^n(x):n\ge 0\}}=X$)

A continuous map on a nonempty topological space is point-transitive if it has a dense forward orbit. This is one convention for topological transitivity, but on spaces with isolated points it must be distinguished from the open-set intersection definition. On a nonempty compact metric space, the positive-time open-set intersection property implies point transitivity: for each member $U_j$ of a countable basis of a topology, ${\bigcup }_{n\ge 1}{f}^{-n}(U_j)$ is open and dense, and the Baire category theorem supplies a point in their intersection. Conversely, a surjective point-transitive continuous map has every tail of its dense orbit dense, since $f^k(X)=X$ and continuity sends its orbit closure into the closure of that tail. The map $f(1/j)=1/(j+1)$, $f(0)=0$ on $\{0\}\cup \{1/j:j\ge 1\}$ is point-transitive but fails the open-set intersection property.

An orbit is a dense orbit in a topological space $X$ if its closure is all of $X$. For a forward map orbit, this means that every nonempty open subset contains some iterate $T^{n}x$. A dense orbit visits every region distinguishable by the topology; it need not pass through every point.

An irrational rotation of the circle has dense orbits, whereas a rational rotation has finite periodic orbits. Density is a topological property and does not specify frequencies of visits. Equidistribution and ergodicity make stronger statements involving a specified measure, so neither should be inferred merely from one dense trajectory.