Sub-actions for Anosov flows

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