Measure-preserving flow

A measure-preserving flow on a probability space is a jointly measurable family $\Phi^t:X\to X$ , $t\in\mathbb R$ , satisfying

\Phi^0=\mathrm{id},\qquad \Phi^{s+t}=\Phi^s\Phi^t,\qquad (\Phi^t)_*\mu=\mu.

It is a measure-preserving action of $(\mathbb R,+)$ . Ergodicity of the flow concerns sets invariant under every $\Phi^t$ .

Time-one maps

An ergodic flow can have a nonergodic time-one map. Translation by all real times on $\mathbb R/\mathbb Z$ is ergodic, but its time-one map is the identity.

Smooth models

A smooth flow is generated by a vector field. With an invariant probability measure and strong continuity, its Koopman generator differentiates observables along that field.