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Lyapunov Exponents versus Integrability in Random Conservative Dynamics

Gianluigi Del Magno, João Lopes Dias, José Pedro Gaivão

TL;DR

This work establishes an invariance principle for random, volume-preserving maps on $\mathbb{T}^2$ showing that vanishing Lyapunov exponents force the existence of a measurable invariant structure on the projective bundle $P M$. The authors apply this principle to two fundamental systems: random additive perturbations of convex billiards on constant-curvature surfaces and random perturbations of the standard map. They prove a sharp rigidity result: for random billiards, $\lambda^+ = 0$ almost everywhere if and only if the table is a geodesic disk; for random standard maps, $\lambda^+ = 0$ almost everywhere if and only if the map is integrable ($K=0$). These findings link stochastic perturbations to deterministic integrability, illustrating that randomness preserves integrable structure and that non-integrability manifests as positive Lyapunov exponents under random perturbations.

Abstract

We consider random dynamical systems generated by volume-preserving piecewise $C^{1}$ maps. For this class of random systems, we establish an invariance principle asserting that if the Lyapunov exponents vanish, then there exists a measurable family of probability measures on the projective bundle that is invariant under the projective cocycle induced by the derivative. We apply this principle to two classes of random systems. First, we study random additive perturbations of a single billiard map associated with a strictly convex planar table on a surface of constant curvature. In this setting, we prove that the Lyapunov exponents vanish almost everywhere if and only if the billiard table is a geodesic disk. Second, we consider random additive perturbations of a single standard map and show that the Lyapunov exponents vanish almost everywhere if and only if the standard map is integrable.

Lyapunov Exponents versus Integrability in Random Conservative Dynamics

TL;DR

This work establishes an invariance principle for random, volume-preserving maps on showing that vanishing Lyapunov exponents force the existence of a measurable invariant structure on the projective bundle . The authors apply this principle to two fundamental systems: random additive perturbations of convex billiards on constant-curvature surfaces and random perturbations of the standard map. They prove a sharp rigidity result: for random billiards, almost everywhere if and only if the table is a geodesic disk; for random standard maps, almost everywhere if and only if the map is integrable (). These findings link stochastic perturbations to deterministic integrability, illustrating that randomness preserves integrable structure and that non-integrability manifests as positive Lyapunov exponents under random perturbations.

Abstract

We consider random dynamical systems generated by volume-preserving piecewise maps. For this class of random systems, we establish an invariance principle asserting that if the Lyapunov exponents vanish, then there exists a measurable family of probability measures on the projective bundle that is invariant under the projective cocycle induced by the derivative. We apply this principle to two classes of random systems. First, we study random additive perturbations of a single billiard map associated with a strictly convex planar table on a surface of constant curvature. In this setting, we prove that the Lyapunov exponents vanish almost everywhere if and only if the billiard table is a geodesic disk. Second, we consider random additive perturbations of a single standard map and show that the Lyapunov exponents vanish almost everywhere if and only if the standard map is integrable.
Paper Structure (18 sections, 19 theorems, 110 equations)

This paper contains 18 sections, 19 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Invariant principle for random maps
  3. Random dynamical systems
  4. Random additive perturbations of toral maps
  5. Invariance principle for random maps
  6. Ledrappier's Theorem
  7. Invariant principle for random maps
  8. When $\nu$ is absolutely continuous
  9. Some ergodic properties of random perturbations of continuous toral maps
  10. Vanishing Lyapunov exponents characterize circular billiards
  11. Convex billiards on surfaces with constant curvature
  12. Billiard map in coordinates $(s,\theta)$
  13. Billiard map in coordinates $(s,r)$
  14. Billiard map on the $2$-torus
  15. Circular billiards
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $D$ be a convex billiard table on a surface $S$ of constant curvature, and let $(X_n)_{n\ge0}$ be an i.i.d. sequence of random variables taking values in $\mathbb T^2$ with distribution that is absolutely continuous with respect to the Lebesgue measure on $\mathbb T^2$ and whose support contains

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • ...and 35 more