Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2015
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| _version_ | 1866908329721200640 |
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| author | Galatolo, Stefano Lucena, Rafael |
| author_facet | Galatolo, Stefano Lucena, Rafael |
| contents | We consider transformations preserving a contracting foliation, such that the associated quotient map satisfies a Lasota-Yorke inequality. We prove that the associated transfer operator, acting on suitable normed spaces, has a spectral gap (on which we have quantitative estimation). As an application we consider Lorenz-like two dimensional maps (piecewise hyperbolic with unbounded contraction and expansion rate): we prove that those systems have a spectral gap and we show a quantitative estimate for their statistical stability. Under deterministic perturbations of the system of size $δ$, the physical measure varies continuously, with a modulus of continuity $O(δ\log δ)$, which is asymptotically optimal for this kind of piecewise smooth maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1507_08191 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps Galatolo, Stefano Lucena, Rafael Dynamical Systems 37A25, 37A10, 37C30, 37D50 We consider transformations preserving a contracting foliation, such that the associated quotient map satisfies a Lasota-Yorke inequality. We prove that the associated transfer operator, acting on suitable normed spaces, has a spectral gap (on which we have quantitative estimation). As an application we consider Lorenz-like two dimensional maps (piecewise hyperbolic with unbounded contraction and expansion rate): we prove that those systems have a spectral gap and we show a quantitative estimate for their statistical stability. Under deterministic perturbations of the system of size $δ$, the physical measure varies continuously, with a modulus of continuity $O(δ\log δ)$, which is asymptotically optimal for this kind of piecewise smooth maps. |
| title | Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps |
| topic | Dynamical Systems 37A25, 37A10, 37C30, 37D50 |
| url | https://arxiv.org/abs/1507.08191 |