Orbit closure ($\overline{\{T^{n}x:n\ge 0\}}$)

The forward orbit closure of $x$ under a continuous map $T$ is the closure of its forward orbit. It is a closed forward-invariant set. If $T$ is invertible, one may instead consider the two-sided orbit closure using $n\in \mathbb{Z}$; the convention should be specified. For a minimal point of an invertible compact system the two closures agree.