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Deformations of the Standard Map with Prescribed Actions and Lyapunov Exponents

Yunzhe Li

TL;DR

<3-5 sentence high-level summary> The paper addresses spectral rigidity questions for the Chirikov standard map by constructing analytic deformations that preserve the action or Lyapunov data of infinitely many periodic orbits accumulating on an invariant curve. The authors develop a resonant normal-form framework around a Liouville rotation-number invariant curve and implement a Picard fixed-point scheme to prescribe the resonant Fourier data of the map, thereby achieving deformations with prescribed spectral characteristics. They prove existence of a real-analytic family of standard maps with infinitely many (p_j,q_j)-periodic orbits whose actions or Lyapunov exponents remain constant across the deformation, revealing a form of partially length-isospectral nonrigidity in this conservative setting. The results connect to Mather’s beta function and the length-spectrum analogies in convex billiards, highlighting subtle arithmetic conditions (Liouville-type) that govern the rigidity or nonrigidity of spectral data in dynamical systems.

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.

Deformations of the Standard Map with Prescribed Actions and Lyapunov Exponents

TL;DR

<3-5 sentence high-level summary> The paper addresses spectral rigidity questions for the Chirikov standard map by constructing analytic deformations that preserve the action or Lyapunov data of infinitely many periodic orbits accumulating on an invariant curve. The authors develop a resonant normal-form framework around a Liouville rotation-number invariant curve and implement a Picard fixed-point scheme to prescribe the resonant Fourier data of the map, thereby achieving deformations with prescribed spectral characteristics. They prove existence of a real-analytic family of standard maps with infinitely many (p_j,q_j)-periodic orbits whose actions or Lyapunov exponents remain constant across the deformation, revealing a form of partially length-isospectral nonrigidity in this conservative setting. The results connect to Mather’s beta function and the length-spectrum analogies in convex billiards, highlighting subtle arithmetic conditions (Liouville-type) that govern the rigidity or nonrigidity of spectral data in dynamical systems.

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.

Paper Structure

This paper contains 50 sections, 28 theorems, 447 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation and background
  3. Laplace spectral rigidity and its dynamical analogues
  4. Symplectic maps and the action spectrum
  5. Length spectrum rigidity
  6. Spectral rigidity of convex billiards
  7. Rigidity by spectral data of accumulating periodic orbits
  8. Mather's beta function
  9. An outline of the proof
  10. Notations and basic definitions
  11. Analytic norms.
  12. Resonant projections in Fourier space.
  13. Exact symplectic maps and actions.
  14. Invariant curves and adapted coordinates
  15. Recovering the potential from an RIC.
  16. ...and 35 more sections

Key Result

Theorem 1.1

There exists a nontrivial analytic family of standard maps $(\mathscr{F}_\tau)_{\tau}$ such that, for each $\tau$, the map $\mathscr{F}_\tau$ has infinitely many dynamically distinct We say that two periodic orbits are dynamically distinct if the orbits do not coincide as sets (they are not just rep

Figures (1)

  • Figure 1: An illustration of $\mathcal{D}_\lambda$.

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 5.1
  • Theorem 6.1
  • Remark
  • Remark
  • Remark
  • Theorem 7.1
  • Remark
  • ...and 47 more