--- OA_place: publisher _id: '22255' abstract: - lang: eng text: "This thesis studies spectral rigidity and nonrigidity phenomena in dynamical systems. The central question is whether a dynamical system can be determined, up to a natural conjugacy, from its spectrum. We consider three related spectra: the length spectrum, the action spectrum, and the Lyapunov spectrum.\r\n\r\nThe first part of the thesis concerns Liouville metrics on the two-dimensional torus. It is a long-standing folklore conjecture that Liouville metrics are the only integrable metrics on the torus. We prove a length-spectral rigidity result for linear conformal deformations of Liouville metrics by exploiting the dynamical properties of the rational tori -- analogues of the resonant convex caustics in billiards. We also establish a complementary classification result showing that marked-length-isospectral Liouville metrics are characterized by rearrangements of the one-dimensional functions appearing in their conformal factors, generalizing a theorem of Abbondandolo-Mazzucchelli. In particular, the second result gives nonrigidity examples within the class of Liouville metrics.\r\n\r\nThe second part of the thesis studies the standard map from the viewpoint of action and Lyapunov spectra. We construct nontrivial deformations of the standard map which preserve the symplectic actions (respectively, the Lyapunov exponents) of infinitely many periodic orbits accumulating on an invariant curve. The proof combines a resonant normal form construction with Picard iteration schemes to obtain a sequence of periodic orbits accumulating on an invariant curve with a Liouville rotation number. Within the resonant normal forms we capture the dependence of these periodic orbits on the resonant Fourier coefficients of the dynamics on the invariant curve and, using the contraction mapping principle, obtain a suitable deformation achieving the prescribed spectral data associated with this sequence of orbits. The result can be viewed as a symplectic twist-map analogue of a length-spectral nonrigidity phenomenon for Riemannian manifolds and convex billiards, and it motivates the existence problem for similar 'partially length-isospectral' deformations of strictly convex billiard tables.\r\n" acknowledged_ssus: - _id: E-Lib - _id: CampIT acknowledgement: "The financial support of the ERC grant SPERIG #885707 is gratefully acknowledged.\r\n" alternative_title: - ISTA Thesis article_processing_charge: No author: - first_name: Yunzhe full_name: Li, Yunzhe id: 41cb05d3-f128-11eb-9611-e4e2b3cfba31 last_name: Li citation: ama: Li Y. Spectral rigidity and nonrigidity of dynamical systems. 2026. doi:10.15479/AT-ISTA-22255 apa: Li, Y. (2026). Spectral rigidity and nonrigidity of dynamical systems. Institute of Science and Technology Austria. https://doi.org/10.15479/AT-ISTA-22255 chicago: Li, Yunzhe. “Spectral Rigidity and Nonrigidity of Dynamical Systems.” Institute of Science and Technology Austria, 2026. https://doi.org/10.15479/AT-ISTA-22255. ieee: Y. Li, “Spectral rigidity and nonrigidity of dynamical systems,” Institute of Science and Technology Austria, 2026. ista: Li Y. 2026. Spectral rigidity and nonrigidity of dynamical systems. Institute of Science and Technology Austria. mla: Li, Yunzhe. Spectral Rigidity and Nonrigidity of Dynamical Systems. Institute of Science and Technology Austria, 2026, doi:10.15479/AT-ISTA-22255. short: Y. Li, Spectral Rigidity and Nonrigidity of Dynamical Systems, Institute of Science and Technology Austria, 2026. corr_author: '1' date_created: 2026-07-08T12:44:31Z date_published: 2026-07-11T00:00:00Z date_updated: 2026-07-20T14:58:23Z day: '11' ddc: - '515' degree_awarded: PhD department: - _id: GradSch - _id: VaKa doi: 10.15479/AT-ISTA-22255 doi_confirm: '1' ec_funded: 1 file: - access_level: open_access checksum: 8201cb5a427656a41828ecde8a85c04b content_type: application/pdf creator: yli date_created: 2026-07-14T10:48:45Z date_updated: 2026-07-14T10:48:45Z file_id: '22337' file_name: 2026_Li_Yunzhe_Thesis.pdf file_size: 1260717 relation: main_file - access_level: closed checksum: 19ee8461ed77f5b7980fc9461f778b6f content_type: application/x-zip-compressed creator: yli date_created: 2026-07-14T11:07:18Z date_updated: 2026-07-20T14:00:33Z file_id: '22339' file_name: 2026_Li_Yunzhe_Thesis.zip file_size: 418752 relation: source_file file_date_updated: 2026-07-20T14:00:33Z has_accepted_license: '1' language: - iso: eng month: '07' oa: 1 oa_version: Published Version page: '131' project: - _id: 9B8B92DE-BA93-11EA-9121-9846C619BF3A call_identifier: H2020 grant_number: '885707' name: Spectral rigidity and integrability for billiards and geodesic flows publication_identifier: issn: - 2663-337X publication_status: published publisher: Institute of Science and Technology Austria related_material: record: - id: '22340' relation: part_of_dissertation status: public - id: '22341' relation: part_of_dissertation status: public status: public supervisor: - first_name: Vadim full_name: Kaloshin, Vadim id: FE553552-CDE8-11E9-B324-C0EBE5697425 last_name: Kaloshin orcid: 0000-0002-6051-2628 title: Spectral rigidity and nonrigidity of dynamical systems tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: dissertation user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 year: '2026' ... --- OA_place: repository OA_type: green _id: '22340' abstract: - lang: eng text: We show that Laplace isospectral deformations within a conformal class of generic Liouville metrics on the two-dimensional torus that are linear in the deformation parameter are necessarily trivial. Two of the main ingredients in our proof are a noncancellation result for the wave trace and an analysis of the second order variational formula for the energy functional associated to closed geodesics. Noncancellation allows us to detect parts of the length spectrum from the Laplace spectrum and conclude rational integrability for the deformed geodesic flow (Liouville metrics are folklorically conjectured to be the only Riemannian metrics with integrable geodesic flow on the torus). We then use the second variational formula to show how the preservation of a single rational torus is sufficient to conclude triviality of the deformation, assuming linearity. We also present some evidence that our hypothesis of linearity may indeed be necessary. article_processing_charge: No arxiv: 1 author: - first_name: Sven Joscha full_name: Henheik, Sven Joscha id: 31d731d7-d235-11ea-ad11-b50331c8d7fb last_name: Henheik orcid: 0000-0003-1106-327X - first_name: Vadim full_name: Kaloshin, Vadim id: FE553552-CDE8-11E9-B324-C0EBE5697425 last_name: Kaloshin orcid: 0000-0002-6051-2628 - first_name: Yunzhe full_name: Li, Yunzhe id: 41cb05d3-f128-11eb-9611-e4e2b3cfba31 last_name: Li - first_name: Amir full_name: Vig, Amir id: 49d58dd5-45f5-11ec-9f86-8ce1276989b9 last_name: Vig citation: ama: Henheik SJ, Kaloshin V, Li Y, Vig A. Spectral rigidity of Liouville tori. arXiv. doi:10.48550/ARXIV.2511.10398 apa: Henheik, S. J., Kaloshin, V., Li, Y., & Vig, A. (n.d.). Spectral rigidity of Liouville tori. arXiv. https://doi.org/10.48550/ARXIV.2511.10398 chicago: Henheik, Sven Joscha, Vadim Kaloshin, Yunzhe Li, and Amir Vig. “Spectral Rigidity of Liouville Tori.” ArXiv, n.d. https://doi.org/10.48550/ARXIV.2511.10398. ieee: S. J. Henheik, V. Kaloshin, Y. Li, and A. Vig, “Spectral rigidity of Liouville tori,” arXiv. . ista: Henheik SJ, Kaloshin V, Li Y, Vig A. Spectral rigidity of Liouville tori. arXiv, 10.48550/ARXIV.2511.10398. mla: Henheik, Sven Joscha, et al. “Spectral Rigidity of Liouville Tori.” ArXiv, doi:10.48550/ARXIV.2511.10398. short: S.J. Henheik, V. Kaloshin, Y. Li, A. Vig, ArXiv (n.d.). corr_author: '1' date_created: 2026-07-14T12:51:50Z date_published: 2025-11-13T00:00:00Z date_updated: 2026-07-20T14:58:23Z day: '13' department: - _id: VaKa - _id: LaEr doi: 10.48550/ARXIV.2511.10398 external_id: arxiv: - '2511.10398' keyword: - Differential Geometry (math.DG) - Mathematical Physics (math-ph) - Dynamical Systems (math.DS) - Spectral Theory (math.SP) - 'FOS: Mathematics' - 'FOS: Mathematics' - 'FOS: Physical sciences' - 'FOS: Physical sciences' - 58J42 - 37J35 - 37J35 - 35P20 - 58J40 - 58J50 - 37D40 language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2511.10398 month: '11' oa: 1 oa_version: Preprint publication: arXiv publication_status: draft related_material: record: - id: '22255' relation: dissertation_contains status: public status: public title: Spectral rigidity of Liouville tori tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: preprint user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 year: '2025' ... --- OA_place: repository OA_type: green _id: '22341' abstract: - lang: eng text: We construct nontrivial deformations of the standard map which preserve the symplectic actions, respectively the Lyapunov exponents, of infinitely many periodic orbits accumulating to an invariant curve. The proof uses a resonant normal-form construction to obtain a sequence of periodic orbits accumulating on an invariant curve with a Liouville rotation number. Within these normal forms we capture the dependence of these periodic orbits on the resonant Fourier coefficients of the dynamics on the invariant curve and, using the contraction mapping principle, obtain a suitable deformation achieving the prescribed spectral data associated with this sequence of orbits. The result can be viewed as a symplectic twist-map analogue of a length spectral nonrigidity phenomenon for Riemannian manifolds and convex billiards, and it motivates the existence problem for similar 'partially length-isospectral' deformations of strictly convex billiard tables. acknowledgement: " The author would like to thank Vadim Kaloshin for initiating this project and for his guidance throughout its development. The author also\r\nthanks Abed Bounemoura, Bassam Fayad, Mathieu Helfter, Comlan Edmond Koudjinan, Illya Koval, Yi Pan, and Daniel Tsodikovich for many helpful discussions. The\r\nfinancial support of the ERC grant SPERIG #885707 is gratefully acknowledged." article_processing_charge: No arxiv: 1 author: - first_name: Yunzhe full_name: Li, Yunzhe id: 41cb05d3-f128-11eb-9611-e4e2b3cfba31 last_name: Li citation: ama: Li Y. Deformations of the standard map with prescribed actions and Lyapunov exponents. arXiv. doi:10.48550/ARXIV.2512.03865 apa: Li, Y. (n.d.). Deformations of the standard map with prescribed actions and Lyapunov exponents. arXiv. https://doi.org/10.48550/ARXIV.2512.03865 chicago: Li, Yunzhe. “Deformations of the Standard Map with Prescribed Actions and Lyapunov Exponents.” ArXiv, n.d. https://doi.org/10.48550/ARXIV.2512.03865. ieee: Y. Li, “Deformations of the standard map with prescribed actions and Lyapunov exponents,” arXiv. . ista: Li Y. Deformations of the standard map with prescribed actions and Lyapunov exponents. arXiv, 10.48550/ARXIV.2512.03865. mla: Li, Yunzhe. “Deformations of the Standard Map with Prescribed Actions and Lyapunov Exponents.” ArXiv, doi:10.48550/ARXIV.2512.03865. short: Y. Li, ArXiv (n.d.). corr_author: '1' date_created: 2026-07-14T13:02:29Z date_published: 2025-12-03T00:00:00Z date_updated: 2026-07-20T14:58:23Z day: '03' department: - _id: VaKa doi: 10.48550/ARXIV.2512.03865 ec_funded: 1 external_id: arxiv: - '2512.03865' keyword: - Dynamical Systems (math.DS) - 'FOS: Mathematics' - 'FOS: Mathematics' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2512.03865 month: '12' oa: 1 oa_version: Preprint project: - _id: 9B8B92DE-BA93-11EA-9121-9846C619BF3A call_identifier: H2020 grant_number: '885707' name: Spectral rigidity and integrability for billiards and geodesic flows publication: arXiv publication_status: draft related_material: record: - id: '22255' relation: dissertation_contains status: public status: public title: Deformations of the standard map with prescribed actions and Lyapunov exponents type: preprint user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 year: '2025' ...