Sub-actions for Anosov flows

Abstract

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