Subcohomology and a Livsic Theorem for Zooming Systems

Abstract

In the context of continuous zooming systems $f:M \to M$ on a compact metric space $M$, which include the non-uniformly expanding ones, possibly with the presence of a critical set, with the zooming set dense in $M$, we prove that any Hölder potential $ϕ: M \to \mathbb{R}$ for which the integrals $\int ϕdμ\geq 0$ with respect to any $f$-invariant probability $μ$, admits a continuous function $λ_{0} : M \to \mathbb{R}$ (which can be Hölder if some integral is positive) such that \[ ϕ\geq λ_{0}- λ_{0} \circ f. \] This extends a result in [9] for $C^{1}$-expanding maps on the circle $\mathbb{T} = \mathbb{R}/\mathbb{Z}$ to important classes of maps as uniformly expanding, local diffeomorphisms with non-uniform expansion, Viana maps, Benedicks-Carleson maps and Rovella maps. We also give an example beyond the exponential contractions context. Moreover, in the case of the integrals $\int ϕdμ= 0$ with respect to any $f$-invariant probability $μ$ and the set of periodic points to be dense in $M$, we obtain a version of the Livsic Theorem, that is, the functions $λ_{0}$ can be taken such that \[ ϕ= λ_{0}- λ_{0} \circ f. \] Additionally, we also prove that the measure which maximizes the integrals is unique for a residual set of potentials.