---
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:<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>.
  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. <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>.
  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:
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  date_created: 2026-07-14T10:48:45Z
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has_accepted_license: '1'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
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:
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    relation: part_of_dissertation
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  - id: '22341'
    relation: part_of_dissertation
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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:
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  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'
...