Measure rigidity and disintegration: time-one maps of flows

An invariant measure for a flow is, of course, an invariant measure for any of its time-t maps. But the converse is far from being true. Hence, one may naturally ask: What is the obstruction for an invariant measure for the time-one map to be invariant for the flow itself? We give an answer in terms of measure disintegration. Surprisingly all it takes is the measure not to be “too much pathological in the orbits”. We prove the following rigidity result. If $\mu$ is an ergodic probability for the time-one map of a flow, then it is either highly pathological in the orbits, or it is highly regular (i.e invariant for the flow). In

particular this measure rigidity result is also true for measurable flows by the classical Ambrose-Kakutani’s representation theorem for measurable flows.