Linear response due to singularities
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866916166789758976 |
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| author | Bahsoun, Wael Galatolo, Stefano |
| author_facet | Bahsoun, Wael Galatolo, Stefano |
| contents | It is well known that a family of tent-like maps with bounded derivatives has no linear response for typical deterministic perturbations changing the value of the turning point. In this note we prove the following result: if we consider a tent-like family with a \emph{cusp} at the turning point, we recover the linear response.
More precisely, let $T_\eps$ be a family of such cusp maps generated by changing the value of the turning point of $T_0$ by a deterministic perturbation and let $h_\eps$ be the corresponding invariant density. We prove that $\eps\mapsto h_\eps$ is differentiable in $L^1$ and provide a formula for its derivative. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_02301 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Linear response due to singularities Bahsoun, Wael Galatolo, Stefano Dynamical Systems Chaotic Dynamics 37A05, 37E05 It is well known that a family of tent-like maps with bounded derivatives has no linear response for typical deterministic perturbations changing the value of the turning point. In this note we prove the following result: if we consider a tent-like family with a \emph{cusp} at the turning point, we recover the linear response. More precisely, let $T_\eps$ be a family of such cusp maps generated by changing the value of the turning point of $T_0$ by a deterministic perturbation and let $h_\eps$ be the corresponding invariant density. We prove that $\eps\mapsto h_\eps$ is differentiable in $L^1$ and provide a formula for its derivative. |
| title | Linear response due to singularities |
| topic | Dynamical Systems Chaotic Dynamics 37A05, 37E05 |
| url | https://arxiv.org/abs/2301.02301 |