2022/08/28 by Mbarki, Lamine, Santana, Eduardo
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2208.13209
In the context of continuous zooming systems f:M → 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 → ℝ for which the integrals ∫ ϕdμ≥ 0 with respect to any f-invariant probability μ, admits a continuous function λ0 : M → ℝ (which can be Hölder if some integral is positive) such that ϕ≥ λ0- λ0 ∘ f. This extends a result in [9] for C1-expanding maps on the circle \mathbbT = 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. Moreover, in the case of the integrals ∫ 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 λ0 can be taken such that ϕ= λ0- λ0 ∘ f. Additionally, we also prove that the measure which maximizes the integrals is unique for a residual set of potentials.