Deformations of the Standard Map with Prescribed Actions and Lyapunov Exponents

TL;DR

This paper develops a method to deform the standard map while preserving specific dynamical features like actions and Lyapunov exponents of many periodic orbits, revealing new insights into symplectic twist maps.

Contribution

It introduces a resonant normal-form construction to create deformations of the standard map with prescribed spectral data for infinitely many periodic orbits.

Findings

Constructed deformations preserve actions and Lyapunov exponents of orbits

Established a normal-form approach for controlling spectral properties

Linked the results to length spectral phenomena in billiards

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.