Title  Measure rigidity and disintegration: timeone maps of flows 
Publication Type  Miscellaneous 
Year of Publication  2017 
Authors  Ponce G, Varao R 
Abstract  An invariant measure for a flow is, of course, an invariant measure for any of its timet maps. But the converse is far from being true. Hence, one may naturally ask: What is the obstruction for an invariant measure for the timeone 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 timeone 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 AmbroseKakutani’s representation theorem for measurable flows.

URL  https://arxiv.org/pdf/1706.00044.pdf 
Citation Key  PonceVarao 