[{"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"ec_funded":1,"alternative_title":["ISTA Thesis"],"status":"public","day":"11","citation":{"mla":"Li, Yunzhe. <i>Spectral Rigidity and Nonrigidity of Dynamical Systems</i>. Institute of Science and Technology Austria, 2026, doi:<a href=\"https://doi.org/10.15479/AT-ISTA-22255\">10.15479/AT-ISTA-22255</a>.","apa":"Li, Y. (2026). <i>Spectral rigidity and nonrigidity of dynamical systems</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT-ISTA-22255\">https://doi.org/10.15479/AT-ISTA-22255</a>","chicago":"Li, Yunzhe. “Spectral Rigidity and Nonrigidity of Dynamical Systems.” Institute of Science and Technology Austria, 2026. <a href=\"https://doi.org/10.15479/AT-ISTA-22255\">https://doi.org/10.15479/AT-ISTA-22255</a>.","ama":"Li Y. Spectral rigidity and nonrigidity of dynamical systems. 2026. doi:<a href=\"https://doi.org/10.15479/AT-ISTA-22255\">10.15479/AT-ISTA-22255</a>","ista":"Li Y. 2026. Spectral rigidity and nonrigidity of dynamical systems. Institute of Science and Technology Austria.","short":"Y. Li, Spectral Rigidity and Nonrigidity of Dynamical Systems, Institute of Science and Technology Austria, 2026.","ieee":"Y. Li, “Spectral rigidity and nonrigidity of dynamical systems,” Institute of Science and Technology Austria, 2026."},"project":[{"_id":"9B8B92DE-BA93-11EA-9121-9846C619BF3A","grant_number":"885707","call_identifier":"H2020","name":"Spectral rigidity and integrability for billiards and geodesic flows"}],"oa_version":"Published Version","type":"dissertation","acknowledgement":"The financial support of the ERC grant SPERIG #885707 is gratefully acknowledged.\r\n","_id":"22255","file_date_updated":"2026-07-20T14:00:33Z","oa":1,"date_published":"2026-07-11T00:00:00Z","month":"07","language":[{"iso":"eng"}],"date_updated":"2026-07-20T14:58:23Z","publisher":"Institute of Science and Technology Austria","department":[{"_id":"GradSch"},{"_id":"VaKa"}],"date_created":"2026-07-08T12:44:31Z","year":"2026","doi":"10.15479/AT-ISTA-22255","publication_identifier":{"issn":["2663-337X"]},"corr_author":"1","file":[{"file_id":"22337","date_updated":"2026-07-14T10:48:45Z","file_size":1260717,"relation":"main_file","checksum":"8201cb5a427656a41828ecde8a85c04b","access_level":"open_access","content_type":"application/pdf","file_name":"2026_Li_Yunzhe_Thesis.pdf","date_created":"2026-07-14T10:48:45Z","creator":"yli"},{"creator":"yli","date_created":"2026-07-14T11:07:18Z","content_type":"application/x-zip-compressed","file_name":"2026_Li_Yunzhe_Thesis.zip","access_level":"closed","checksum":"19ee8461ed77f5b7980fc9461f778b6f","relation":"source_file","date_updated":"2026-07-20T14:00:33Z","file_size":418752,"file_id":"22339"}],"has_accepted_license":"1","ddc":["515"],"author":[{"first_name":"Yunzhe","last_name":"Li","full_name":"Li, Yunzhe","id":"41cb05d3-f128-11eb-9611-e4e2b3cfba31"}],"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","OA_place":"publisher","acknowledged_ssus":[{"_id":"E-Lib"},{"_id":"CampIT"}],"related_material":{"record":[{"status":"public","id":"22340","relation":"part_of_dissertation"},{"status":"public","relation":"part_of_dissertation","id":"22341"}]},"article_processing_charge":"No","page":"131","publication_status":"published","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"}],"doi_confirm":"1","supervisor":[{"full_name":"Kaloshin, Vadim","id":"FE553552-CDE8-11E9-B324-C0EBE5697425","orcid":"0000-0002-6051-2628","last_name":"Kaloshin","first_name":"Vadim"}],"degree_awarded":"PhD","title":"Spectral rigidity and nonrigidity of dynamical systems"},{"article_processing_charge":"No","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."}],"external_id":{"arxiv":["2511.10398"]},"publication_status":"draft","title":"Spectral rigidity of Liouville tori","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"],"author":[{"full_name":"Henheik, Sven Joscha","id":"31d731d7-d235-11ea-ad11-b50331c8d7fb","orcid":"0000-0003-1106-327X","last_name":"Henheik","first_name":"Sven Joscha"},{"last_name":"Kaloshin","first_name":"Vadim","full_name":"Kaloshin, Vadim","id":"FE553552-CDE8-11E9-B324-C0EBE5697425","orcid":"0000-0002-6051-2628"},{"id":"41cb05d3-f128-11eb-9611-e4e2b3cfba31","full_name":"Li, Yunzhe","last_name":"Li","first_name":"Yunzhe"},{"last_name":"Vig","first_name":"Amir","full_name":"Vig, Amir","id":"49d58dd5-45f5-11ec-9f86-8ce1276989b9"}],"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","OA_place":"repository","related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"22255"}]},"corr_author":"1","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2511.10398","open_access":"1"}],"year":"2025","date_created":"2026-07-14T12:51:50Z","doi":"10.48550/ARXIV.2511.10398","publication":"arXiv","oa":1,"language":[{"iso":"eng"}],"month":"11","date_published":"2025-11-13T00:00:00Z","arxiv":1,"department":[{"_id":"VaKa"},{"_id":"LaEr"}],"date_updated":"2026-07-20T14:58:23Z","oa_version":"Preprint","type":"preprint","_id":"22340","citation":{"mla":"Henheik, Sven Joscha, et al. “Spectral Rigidity of Liouville Tori.” <i>ArXiv</i>, doi:<a href=\"https://doi.org/10.48550/ARXIV.2511.10398\">10.48550/ARXIV.2511.10398</a>.","chicago":"Henheik, Sven Joscha, Vadim Kaloshin, Yunzhe Li, and Amir Vig. “Spectral Rigidity of Liouville Tori.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/ARXIV.2511.10398\">https://doi.org/10.48550/ARXIV.2511.10398</a>.","apa":"Henheik, S. J., Kaloshin, V., Li, Y., &#38; Vig, A. (n.d.). Spectral rigidity of Liouville tori. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/ARXIV.2511.10398\">https://doi.org/10.48550/ARXIV.2511.10398</a>","ama":"Henheik SJ, Kaloshin V, Li Y, Vig A. Spectral rigidity of Liouville tori. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/ARXIV.2511.10398\">10.48550/ARXIV.2511.10398</a>","ista":"Henheik SJ, Kaloshin V, Li Y, Vig A. Spectral rigidity of Liouville tori. arXiv, <a href=\"https://doi.org/10.48550/ARXIV.2511.10398\">10.48550/ARXIV.2511.10398</a>.","short":"S.J. Henheik, V. Kaloshin, Y. Li, A. Vig, ArXiv (n.d.).","ieee":"S. J. Henheik, V. Kaloshin, Y. Li, and A. Vig, “Spectral rigidity of Liouville tori,” <i>arXiv</i>. ."},"OA_type":"green","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"status":"public","day":"13"},{"doi":"10.48550/ARXIV.2512.03865","publication":"arXiv","year":"2025","date_created":"2026-07-14T13:02:29Z","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2512.03865","open_access":"1"}],"corr_author":"1","related_material":{"record":[{"id":"22255","relation":"dissertation_contains","status":"public"}]},"author":[{"full_name":"Li, Yunzhe","id":"41cb05d3-f128-11eb-9611-e4e2b3cfba31","first_name":"Yunzhe","last_name":"Li"}],"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","OA_place":"repository","title":"Deformations of the standard map with prescribed actions and Lyapunov exponents","keyword":["Dynamical Systems (math.DS)","FOS: Mathematics","FOS: Mathematics"],"article_processing_charge":"No","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."}],"publication_status":"draft","external_id":{"arxiv":["2512.03865"]},"status":"public","day":"03","ec_funded":1,"OA_type":"green","citation":{"short":"Y. Li, ArXiv (n.d.).","ieee":"Y. Li, “Deformations of the standard map with prescribed actions and Lyapunov exponents,” <i>arXiv</i>. .","ama":"Li Y. Deformations of the standard map with prescribed actions and Lyapunov exponents. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/ARXIV.2512.03865\">10.48550/ARXIV.2512.03865</a>","ista":"Li Y. Deformations of the standard map with prescribed actions and Lyapunov exponents. arXiv, <a href=\"https://doi.org/10.48550/ARXIV.2512.03865\">10.48550/ARXIV.2512.03865</a>.","apa":"Li, Y. (n.d.). Deformations of the standard map with prescribed actions and Lyapunov exponents. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/ARXIV.2512.03865\">https://doi.org/10.48550/ARXIV.2512.03865</a>","chicago":"Li, Yunzhe. “Deformations of the Standard Map with Prescribed Actions and Lyapunov Exponents.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/ARXIV.2512.03865\">https://doi.org/10.48550/ARXIV.2512.03865</a>.","mla":"Li, Yunzhe. “Deformations of the Standard Map with Prescribed Actions and Lyapunov Exponents.” <i>ArXiv</i>, doi:<a href=\"https://doi.org/10.48550/ARXIV.2512.03865\">10.48550/ARXIV.2512.03865</a>."},"project":[{"call_identifier":"H2020","name":"Spectral rigidity and integrability for billiards and geodesic flows","_id":"9B8B92DE-BA93-11EA-9121-9846C619BF3A","grant_number":"885707"}],"_id":"22341","oa_version":"Preprint","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.","type":"preprint","language":[{"iso":"eng"}],"month":"12","date_published":"2025-12-03T00:00:00Z","arxiv":1,"department":[{"_id":"VaKa"}],"date_updated":"2026-07-20T14:58:23Z","oa":1}]