The convergence-divergence criterion for the Mañé cohomology lemma and Livšic theorem

Abstract

We establish a new dynamical convergence-divergence criterion for the existence of Lipschitz functions with continuous sub-actions, over general dynamical systems. This unifies and generalises various versions of the Mañé cohomology lemma, in the setting of uniformly expanding maps, and systems with indifferent fixed points. One consequence is that for any compact metric space $(X,ρ)$, and Lipschitz Ruelle-expanding map $T:X\to X$, a metric transform $ω$ satisfies Dini's criterion $\int_0 ω(s)s^{-1}\, ds <\infty$ if and only if every function that is Lipschitz for $ω\circρ$ has a continuous sub-action. Another consequence is that if $T$ is an expanding circle map of Manneville-Pomeau type $α\in(0,1)$, then $ω$ satisfies the convergence criterion $\int_0 ω(s)s^{-1-α}\, ds <\infty$ if and only if every function that is Lipschitz for $ω\circρ$ has a continuous sub-action. For the latter class of maps $T$, with $ω$ satisfying the convergence criterion, we prove a Livšic theorem for functions that are Lipschitz for $ω\circρ$: if $\int f\, dμ=0$ for all invariant measures $μ$, then there exists a continuous function $φ$ such that $f=φ\circ T-φ$, where $φ$ is Lipschitz with respect to $D_α(ω)\circρ$, and $D_α(ω)(t)= \int_0^t ω(s)s^{-1-α} ds$.