Certified spectral approximation of transfer operators and the Gauss map
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914361121964032 |
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| author | Nisoli, Isaia |
| author_facet | Nisoli, Isaia |
| contents | We prove that the full spectral picture of a transfer operator (every isolated eigenvalue, eigenvector, and Riesz projector outside the essential spectral radius) can be approximated to arbitrary precision by finite-rank discretizations, with no spectral pollution. The method is a~posteriori: once a computable approximation bound is available (from compactness or a Doeblin--Fortet--Lasota--Yorke inequality, which may require hyperbolicity constants and adapted Banach spaces), the spectral gap and multiplicity are certified from computed data via a resolvent perturbation bound, with no further dynamical input. This applies both to compact operators on a single Banach space and to quasi-compact operators satisfying a Doeblin--Fortet--Lasota--Yorke inequality, extending Li's resolution of the Ulam conjecture from the invariant density to the entire discrete spectrum with certified error bounds at every finite truncation.
As a benchmark, we certify the first $50$ nonzero eigenvalues of the Gauss--Kuzmin--Wirsing operator to at least $90$ rigorous decimal digits, together with their eigenvectors, Riesz projectors, and spectral gap, yielding a certified spectral expansion for Gauss--Kuzmin distributions with explicit error bounds and providing a rigorous answer to the Gauss--Babenko--Knuth problem on the spectral data of the Gauss map. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_19435 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Certified spectral approximation of transfer operators and the Gauss map Nisoli, Isaia Dynamical Systems Functional Analysis Primary: 37C30, 47A75. Secondary: 37M25, 65G20, 11K50, 47A58 We prove that the full spectral picture of a transfer operator (every isolated eigenvalue, eigenvector, and Riesz projector outside the essential spectral radius) can be approximated to arbitrary precision by finite-rank discretizations, with no spectral pollution. The method is a~posteriori: once a computable approximation bound is available (from compactness or a Doeblin--Fortet--Lasota--Yorke inequality, which may require hyperbolicity constants and adapted Banach spaces), the spectral gap and multiplicity are certified from computed data via a resolvent perturbation bound, with no further dynamical input. This applies both to compact operators on a single Banach space and to quasi-compact operators satisfying a Doeblin--Fortet--Lasota--Yorke inequality, extending Li's resolution of the Ulam conjecture from the invariant density to the entire discrete spectrum with certified error bounds at every finite truncation. As a benchmark, we certify the first $50$ nonzero eigenvalues of the Gauss--Kuzmin--Wirsing operator to at least $90$ rigorous decimal digits, together with their eigenvectors, Riesz projectors, and spectral gap, yielding a certified spectral expansion for Gauss--Kuzmin distributions with explicit error bounds and providing a rigorous answer to the Gauss--Babenko--Knuth problem on the spectral data of the Gauss map. |
| title | Certified spectral approximation of transfer operators and the Gauss map |
| topic | Dynamical Systems Functional Analysis Primary: 37C30, 47A75. Secondary: 37M25, 65G20, 11K50, 47A58 |
| url | https://arxiv.org/abs/2602.19435 |