In the context of zooming maps f:M→M on a compact metric space M,
which include the non-uniformly expanding ones, possibly with the presence of a
critical set, if the map is topologically exact, we prove that a potential
ϕ:M→R for which the integrals ∫ϕdμ≥0 with
respect to any f-invariant probability μ, admits a coboundary
λ0​−λ0​∘T such that ϕ≥λ0​−λ0​∘T. This extends a result for C1-expanding maps on the
circle T=R/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.Comment: This new version contains substantial changes, as a new proof for the
main theorem. arXiv admin note: substantial text overlap with
arXiv:2010.0814