Deformations of the standard map with prescribed actions and Lyapunov exponents
Li Y. Deformations of the standard map with prescribed actions and Lyapunov exponents. arXiv, 10.48550/ARXIV.2512.03865.
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Abstract
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.
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2025-12-03
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arXiv
Acknowledgement
The author would like to thank Vadim Kaloshin for initiating this project and for his guidance throughout its development. The author also
thanks Abed Bounemoura, Bassam Fayad, Mathieu Helfter, Comlan Edmond Koudjinan, Illya Koval, Yi Pan, and Daniel Tsodikovich for many helpful discussions. The
financial support of the ERC grant SPERIG #885707 is gratefully acknowledged.
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Cite this
Li Y. Deformations of the standard map with prescribed actions and Lyapunov exponents. arXiv. doi:10.48550/ARXIV.2512.03865
Li, Y. (n.d.). Deformations of the standard map with prescribed actions and Lyapunov exponents. arXiv. https://doi.org/10.48550/ARXIV.2512.03865
Li, Yunzhe. “Deformations of the Standard Map with Prescribed Actions and Lyapunov Exponents.” ArXiv, n.d. https://doi.org/10.48550/ARXIV.2512.03865.
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, 10.48550/ARXIV.2512.03865.
Li, Yunzhe. “Deformations of the Standard Map with Prescribed Actions and Lyapunov Exponents.” ArXiv, doi:10.48550/ARXIV.2512.03865.
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