Deformations of the standard map with prescribed actions and Lyapunov exponents

preprint Deformations of the standard map with prescribed actions and Lyapunov exponents draft Yunzhe Li author 41cb05d3-f128-11eb-9611-e4e2b3cfba31 VaKa department Spectral rigidity and integrability for billiards and geodesic flows project 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. 2025 eng Dynamical Systems (math.DS)FOS: MathematicsFOS: Mathematics arXiv 2512.0386510.48550/ARXIV.2512.03865 https://research-explorer.ista.ac.at/record/22255 Li, Yunzhe. “Deformations of the Standard Map with Prescribed Actions and Lyapunov Exponents.” ArXiv, n.d. <a href="https://doi.org/10.48550/ARXIV.2512.03865">https://doi.org/10.48550/ARXIV.2512.03865</a>. Y. Li, “Deformations of the standard map with prescribed actions and Lyapunov exponents,” arXiv. . Li Y. Deformations of the standard map with prescribed actions and Lyapunov exponents. arXiv. doi:<a href="https://doi.org/10.48550/ARXIV.2512.03865">10.48550/ARXIV.2512.03865</a> Li, Y. (n.d.). Deformations of the standard map with prescribed actions and Lyapunov exponents. arXiv. <a href="https://doi.org/10.48550/ARXIV.2512.03865">https://doi.org/10.48550/ARXIV.2512.03865</a> 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>. Y. Li, ArXiv (n.d.). Li, Yunzhe. “Deformations of the Standard Map with Prescribed Actions and Lyapunov Exponents.” ArXiv, doi:<a href="https://doi.org/10.48550/ARXIV.2512.03865">10.48550/ARXIV.2512.03865</a>. 223412026-07-14T13:02:29Z2026-07-20T14:58:23Z