---
res:
  bibo_abstract:
  - "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@eng"
  bibo_authorlist:
  - foaf_Person:
      foaf_givenName: Yunzhe
      foaf_name: Li, Yunzhe
      foaf_surname: Li
      foaf_workInfoHomepage: http://www.librecat.org/personId=41cb05d3-f128-11eb-9611-e4e2b3cfba31
  bibo_doi: 10.15479/AT-ISTA-22255
  dct_date: 2026^xs_gYear
  dct_isPartOf:
  - http://id.crossref.org/issn/2663-337X
  dct_language: eng
  dct_publisher: Institute of Science and Technology Austria@
  dct_title: Spectral rigidity and nonrigidity of dynamical systems@
...