= Dense orbit
{title2=$\overline{\mathcal O(x)}=X$}
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 $T^nx$. 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.
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