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 CR(ϕ) and has the invariantly measure shadowing property on CR(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 CR(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.