---
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. <i>arXiv</i>. doi:<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>.
  ieee: Y. Li, “Deformations of the standard map with prescribed actions and Lyapunov
    exponents,” <i>arXiv</i>. .
  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>.
  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>.
  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'
...