# Measure Rigidity and Disintegration: Time-one map of flows

## TL;DR

This paper investigates when an invariant measure for a time-one map of a flow is also invariant for the flow itself, revealing a measure disintegration criterion that distinguishes regular from pathological measures.

## Contribution

It establishes a measure rigidity theorem linking invariance of the flow to properties of the measure for the time-one map, using measure disintegration techniques.

## Key findings

- Ergodic measures for the time-one map are either highly regular or highly pathological.
- The measure rigidity result extends to measurable flows via Ambrose-Kakutani's theorem.
- The key criterion involves the measure's disintegration along orbits.

## Abstract

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.