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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| Accesso online: | https://arxiv.org/abs/2208.13209 |
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| _version_ | 1866915242038001664 |
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| author | Mbarki, Lamine Santana, Eduardo |
| author_facet | Mbarki, Lamine Santana, Eduardo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_13209 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Subcohomology and a Livsic Theorem for Zooming Systems Mbarki, Lamine Santana, Eduardo Dynamical Systems 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. |
| title | Subcohomology and a Livsic Theorem for Zooming Systems |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2208.13209 |