Minimal set

Remarks

Equivalent form: a nonempty closed invariant set $M$ is minimal if and only if, for every $x \in M$, the forward orbit $\set{T^n(x) : n \ge 0}$ is dense in $M$. Indeed, the closure of a forward orbit is closed and invariant by continuity of $T$. If $X$ is nonempty and compact, Zorn's lemma applied to the nonempty closed invariant sets gives at least one minimal set. For a continuous action of a group $G$, a minimal set is a nonempty closed $G$-invariant set with no proper nonempty closed $G$-invariant subset. The action is minimal when every orbit is dense. Example: for $T(x) = x + \alpha \bmod 1$ on the circle $\R / \Z$ with $\alpha$ irrational, the points $n \alpha \bmod 1$ are dense, so every orbit is dense and the circle is minimal. Non-example: for $\alpha = 1/2$, the circle is not minimal, since the closed invariant set $\set{0, 1/2}$ is a proper nonempty subset; clause 3 fails.