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and strong orbit equivalent substitution
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An Ambrose-Kakutani representation theorem for countable-to-1 semiflows
Let $X$ be a Polish space and $T_t$ a
jointly Borel measurable action of $\mathbb{R}^+ = [0, \infty)$ on
$X$ by surjective maps preserving some Borel probability measure
$\mu$ on $X$. We show that if each $T_t$ is countable-to-1 and if
$T_t$ has the "discrete orbit branching property'' (described in
the introduction), then $(X, T_t)$ is isomorphic to a "semiflow
under a function''.