Dense orbit ($\overline{\mathcal{O}(x)}=X$)

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.