Continuity of Conditional Measures Associated to Measure-preserving Semiflows

LetX be a standard probability space and Tt a measure-preserving semiflow on X. We show that there exists a set X0 of full measure in X such that for any x ∈ X0 and t ≥ 0 there are measures μx,t and μ − x,t which for all but a countable number of t give a distribution on the set of points y such that Tt(y) = Tt(x). These measures arise by taking weak∗−limits of suitable conditional expectations. Say that a point x has a measurable orbit discontinuity at time t0 if either μ + x,t or μ − x,t is weak ∗−discontinuous in t at t0. We show that there exists an invariant set of full measure in X such that any point in this set has at most countably many measurable orbit discontinuities. Furthermore we show that if x has a measurable orbit discontinuity at time 0, then x has an orbit discontinuity at time 0 in the sense of Orbit discontinuities and topological models for Bordel semiflows, D. McClendon.