Optimal linear response for Anosov diffeomorphisms
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915996524085248 |
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| author | Froyland, Gary Phalempin, Maxence |
| author_facet | Froyland, Gary Phalempin, Maxence |
| contents | It is well known that an Anosov diffeomorphism $T$ enjoys linear response of its SRB measure with respect to infinitesimal perturbations $\dot{T}$. For a fixed observation function $c$, we develop a theory to optimise the response of the SRB-expectation of $c$. Our approach is based on the response of the transfer operator on the anisotropic Banach spaces of Gouëzel--Liverani. We prove that the optimising perturbation $\dot{T}$ is unique for non-degenerate response functions and provide explicit expressions for the Fourier coefficients of $\dot{T}$. We develop an efficient Fourier-based numerical scheme to approximate the optimal vector field $\dot{T}$, along with a proof of convergence. The utility of our approach is illustrated in two numerical examples, by localising SRB measures with small, optimally selected, perturbations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_16532 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal linear response for Anosov diffeomorphisms Froyland, Gary Phalempin, Maxence Dynamical Systems It is well known that an Anosov diffeomorphism $T$ enjoys linear response of its SRB measure with respect to infinitesimal perturbations $\dot{T}$. For a fixed observation function $c$, we develop a theory to optimise the response of the SRB-expectation of $c$. Our approach is based on the response of the transfer operator on the anisotropic Banach spaces of Gouëzel--Liverani. We prove that the optimising perturbation $\dot{T}$ is unique for non-degenerate response functions and provide explicit expressions for the Fourier coefficients of $\dot{T}$. We develop an efficient Fourier-based numerical scheme to approximate the optimal vector field $\dot{T}$, along with a proof of convergence. The utility of our approach is illustrated in two numerical examples, by localising SRB measures with small, optimally selected, perturbations. |
| title | Optimal linear response for Anosov diffeomorphisms |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2504.16532 |