A pseudospectral approach to rigorous numerical estimation of Ruelle resonances of transfer operators
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916840286978048 |
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| author | Blumenthal, Alex Nisoli, Isaia Taylor-Crush, Toby |
| author_facet | Blumenthal, Alex Nisoli, Isaia Taylor-Crush, Toby |
| contents | Resonances, isolated eigenvalues of a transfer operator acting on suitably chosen Banach spaces, play a fundamental role in understanding the statistical properties of chaotic dynamical systems. In this paper, we introduce a pseudospectral approach, inspired by Householder's theorem, for the rigorous, computer-assisted estimation of resonances, providing regions where resonances must exist and precluding the presence of resonances elsewhere. The approach is general, and applies to the transfer operators of a wide variety of chaotic systems, including Anosov/ Axiom A diffeomorphisms and piecewise expanding maps. We implement this approach computationally for a class of analytic uniformly expanding maps of the circle. We anticipate that the pseudospectral framework developed here will be broadly applicable to other spectral problems in dynamical systems and beyond. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_09021 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A pseudospectral approach to rigorous numerical estimation of Ruelle resonances of transfer operators Blumenthal, Alex Nisoli, Isaia Taylor-Crush, Toby Dynamical Systems Functional Analysis Primary: 37C30, Secondary: 47A10, 65G99 Resonances, isolated eigenvalues of a transfer operator acting on suitably chosen Banach spaces, play a fundamental role in understanding the statistical properties of chaotic dynamical systems. In this paper, we introduce a pseudospectral approach, inspired by Householder's theorem, for the rigorous, computer-assisted estimation of resonances, providing regions where resonances must exist and precluding the presence of resonances elsewhere. The approach is general, and applies to the transfer operators of a wide variety of chaotic systems, including Anosov/ Axiom A diffeomorphisms and piecewise expanding maps. We implement this approach computationally for a class of analytic uniformly expanding maps of the circle. We anticipate that the pseudospectral framework developed here will be broadly applicable to other spectral problems in dynamical systems and beyond. |
| title | A pseudospectral approach to rigorous numerical estimation of Ruelle resonances of transfer operators |
| topic | Dynamical Systems Functional Analysis Primary: 37C30, Secondary: 47A10, 65G99 |
| url | https://arxiv.org/abs/2507.09021 |