Spectral decomposition and Ω-stability of flows with expanding measures

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