Sub-actions for Anosov flows

Abstract

Let (M, {Øt }) be a smooth (not necessarily transitive) Anosov flow without fixed points generated by a vector field X(x) = (d/dt)|t=0Øt (x) on a compact manifold M. Let A : M → R be a globally Holder function defined on M. Assume that ∫0T 0 A ◦ Øt (x) dt ≥ 0 for any periodic orbit {Øt (x)}t=T t=0 of period T . Then there exists a H¨older function V : M →R, called a sub-action, smooth in the flow direction, such that A(x) ≥ LXV (x), for all x є M (where LXV = (d/dt)|t=0 V ◦Øt(x) denotes the Lie derivative of V ). If A is Cr then LXV is Cr on any local center-stable manifold.