Spectral decomposition and Ω-stability of flows with expanding measures

In this paper we present a measurable version of the classical spectral decomposition theorem for flows. More precisely, we prove that if a flow phi on a compact metric space X is invariantly measure expanding on its chain recurrent set C R (phi) and has the invariantly measure shadowing property on C R (phi) then phi has the spectral decomposition, i.e. the nonwandering set 0(phi) is decomposed by a disjoint union of finitely many invariant and closed subsets on which phi is topologically transitive. Moreover we show that if phi is invariantly measure expanding on CR(phi) then it is invariantly measure expanding on X . Using this, we characterize the measure expanding flows on a compact C-infinity manifold via the notion of Omega-stability. (C) 2020 Elsevier Inc. All rights reserved.