Let $X$ be a compact Haussdorf space, let $\mu$ be a Borel measure on $X$ with $\mathrm{supp}(\mu)=X$ and let $(\phi_s)_{s\in\mathbb R}$ be a one-parameter group of homeomorphisms which is
- continuous in the sense that $\mathbb R\times X \ni (s,r) \mapsto \sigma_s(x) \in X$ is a continuous map, and
- ergodic in the sense that $\phi_s(\mu)$, $s\in\mathbb R$, is a quasi-equivalent family of measures with the property that every $\phi$-invariant measurable subset is a null or a co-null set.
The ergodic system is called transitive if, for almost all $x\in X$, the orbit $O_x= \{ \phi_s(x) : s\in\mathbb R\}$ is a co-null set and non-transitive (or properly ergodic) if, for almost all $x\in X$, the orbit $O_x= \{ \phi_s(x) : s\in\mathbb R\}$ is a null set.
My question is the following: Denote by $p(x) = \min\{s>0 : \sigma_s(x) = x\}$ the minimal period of a point $x$. In the transitive case, almost all points have the same period. What can we say about $p(x)$ in the non-transitive case? Do almost all points have $p(x)=\infty$? If not, do points $x$ with arbitrarily large periods exist?
