Sub-actions for Anosov flows

Abstract

Let $(M,\{\phi^t\})$ be a smooth (not necessarily transitive) Anosov flow without fixed points generated by a vector field $X(x)=(d/dt)|_{t=0}\phi^t(x)$ on a compact manifold M. Let $A:M\rightarrow\mathbb{R}$ be a globally Hölder function defined on M. Assume that $\int_0^T A\circ\phi^t(x)\,dt\geq0$ for any periodic orbit $\{\phi^t(x)\}_{t=0}^{t=T}$ of period T. Then there exists a Hölder function $V:M\rightarrow \mathbb{R}$, called a sub-action, smooth in the flow direction, such that

\[A(x)\geq L_XV(x),\quad\text{for all }x\in M\]

(where $L_XV=(d/dt)|_{t=0}V\circ\phi^t(x)$ denotes the Lie derivative of V). If A is $\mathcal{C}^r$ then L_XV is $\mathcal{C}^r$ on any local center-stable manifold.