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 $\varphi :M\to \mathbb{R}$ for which the integrals $\int \varphi d\mu \ge 0$ with respect to any $f$-invariant probability $\mu$, admits a continuous function ${\lambda }_0:M\to \mathbb{R}$ (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 $\int \varphi d\mu =0$ with respect to any $f$-invariant probability $\mu$ and the set of periodic points to be dense in $M$, we obtain a version of the Livsic Theorem, that is, the functions ${\lambda }_0$ 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.

We give another proof for Lemma 3.1