Quantitative statistical stability for the equilibrium states of piecewise partially hyperbolic maps
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866912338859261952 |
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| author | Bilbao, Rafael Bioni, Ricardo Lucena, Rafael |
| author_facet | Bilbao, Rafael Bioni, Ricardo Lucena, Rafael |
| contents | We consider a class of endomorphisms that contains a set of piecewise partially hyperbolic dynamics semi-conjugated to non-uniformly expanding maps. Our goal is to study a class of endomorphisms that preserve a foliation that is almost everywhere uniformly contracted, with possible discontinuity sets parallel to the contracting direction. We apply the spectral gap property and the $ζ$-Hölder regularity of the disintegration of its equilibrium states to prove a quantitative statistical stability statement. More precisely, under deterministic perturbations of the system of size $δ$, we show that the $F$-invariant measure varies continuously with respect to a suitable anisotropic norm. Moreover, we prove that for certain interesting classes of perturbations, its modulus of continuity is $O(δ^ζ\log δ)$. This article has been accepted for publication in the Discrete and Continuous Dynamical Systems journal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_05679 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Quantitative statistical stability for the equilibrium states of piecewise partially hyperbolic maps Bilbao, Rafael Bioni, Ricardo Lucena, Rafael Dynamical Systems 37A25, 37A10, 37C30, 37D50 We consider a class of endomorphisms that contains a set of piecewise partially hyperbolic dynamics semi-conjugated to non-uniformly expanding maps. Our goal is to study a class of endomorphisms that preserve a foliation that is almost everywhere uniformly contracted, with possible discontinuity sets parallel to the contracting direction. We apply the spectral gap property and the $ζ$-Hölder regularity of the disintegration of its equilibrium states to prove a quantitative statistical stability statement. More precisely, under deterministic perturbations of the system of size $δ$, we show that the $F$-invariant measure varies continuously with respect to a suitable anisotropic norm. Moreover, we prove that for certain interesting classes of perturbations, its modulus of continuity is $O(δ^ζ\log δ)$. This article has been accepted for publication in the Discrete and Continuous Dynamical Systems journal. |
| title | Quantitative statistical stability for the equilibrium states of piecewise partially hyperbolic maps |
| topic | Dynamical Systems 37A25, 37A10, 37C30, 37D50 |
| url | https://arxiv.org/abs/2008.05679 |