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Discontinuity of Lyapunov exponent in spaces of quasiperiodic cocycles: Smoothness vs Arithmetic

Jinhao Liang, Kai Tao, Jiangong You

TL;DR

This work analyzes the continuity of the Lyapunov exponent $L(\alpha,A)$ for quasiperiodic $\mathrm{SL}(2,\mathbb R)$-cocycles across a spectrum of regularity spaces and irrational frequencies. By constructing convergent cocycle sequences and exploiting sharp return-time dynamics tied to the frequency’s continued-fraction data, the authors delineate a precise boundary between continuity and discontinuity: $L(\alpha,A)$ is continuous in Gevrey spaces $G^s$ for $1<s<2$ but admits discontinuities for $s>2$ for strong Diophantine frequencies, with analogous results for $C^\infty$ and other regularity classes. A core innovation is the design of critical intervals and returning-time structures that depend on the arithmetic of the frequency, allowing a transition result that extends beyond bounded-type frequencies to all strong Diophantine frequencies. The paper also provides counterexamples for Brjuno, Diophantine, and Liouvillean regimes, clarifying how arithmetic complexity interacts with smoothness to govern LE continuity, and it extends the framework to Schrödinger cocycles. Overall, the results pinpoint the optimal regularity thresholds (notably $G^2$) governing LE continuity and reveal the delicate balance between arithmetic and analytic structure in quasiperiodic dynamical systems.

Abstract

We construct examples of discontinuity of Lyapunov exponent in the spaces of quasiperiodic $\mathrm{SL}(2,\mathbb R)$-cocycles for fixed irrational frequencies. Especially, we prove that the Gevrey space $G^2$ is the transition space of continuity for all strong Diophantine frequencies. We also construct examples of discontinuity for other frequencies in less smooth spaces, which show that the more difficult it is to approximate the frequency with rational numbers, the more likely it is to exhibit discontinuity in smoother spaces.

Discontinuity of Lyapunov exponent in spaces of quasiperiodic cocycles: Smoothness vs Arithmetic

TL;DR

This work analyzes the continuity of the Lyapunov exponent for quasiperiodic -cocycles across a spectrum of regularity spaces and irrational frequencies. By constructing convergent cocycle sequences and exploiting sharp return-time dynamics tied to the frequency’s continued-fraction data, the authors delineate a precise boundary between continuity and discontinuity: is continuous in Gevrey spaces for but admits discontinuities for for strong Diophantine frequencies, with analogous results for and other regularity classes. A core innovation is the design of critical intervals and returning-time structures that depend on the arithmetic of the frequency, allowing a transition result that extends beyond bounded-type frequencies to all strong Diophantine frequencies. The paper also provides counterexamples for Brjuno, Diophantine, and Liouvillean regimes, clarifying how arithmetic complexity interacts with smoothness to govern LE continuity, and it extends the framework to Schrödinger cocycles. Overall, the results pinpoint the optimal regularity thresholds (notably ) governing LE continuity and reveal the delicate balance between arithmetic and analytic structure in quasiperiodic dynamical systems.

Abstract

We construct examples of discontinuity of Lyapunov exponent in the spaces of quasiperiodic -cocycles for fixed irrational frequencies. Especially, we prove that the Gevrey space is the transition space of continuity for all strong Diophantine frequencies. We also construct examples of discontinuity for other frequencies in less smooth spaces, which show that the more difficult it is to approximate the frequency with rational numbers, the more likely it is to exhibit discontinuity in smoother spaces.

Paper Structure

This paper contains 24 sections, 32 theorems, 394 equations.

Table of Contents

  1. Introduction.
  2. Angles and norms of product of SL(2,$\mathbb R$)-matrices
  3. Sharp estimates of the returning times
  4. Sharp distribution of the trajectory of the irrational shift
  5. Sharp estimates of the returning times for a single interval.
  6. Proof of Theorem \ref{['thm.main1']}: Discontinuity of LE in $C^l$ topology
  7. The lower bound of LE
  8. The upper bound of LE
  9. Proof of Proposition \ref{['prop.lowerbound']} and \ref{['prop.upperbound']}.
  10. Proof of Proposition \ref{['prop.lowerbound']}.
  11. The $N$th step.
  12. The $n$th step.
  13. Proof of Proposition \ref{['prop.upperbound']}.
  14. Proof of Theorem \ref{['thm.main2']}: Discontinuity of LE in $C^\infty$ topology
  15. Proof of Theorem \ref{['thm.main5']} and \ref{['thm.main3']}: Discontinuity of LE in $G^s$ topology
  16. ...and 9 more sections

Key Result

Theorem 1.1

For any fixed $\alpha \in \mathrm{SDC}$, the Gevrey space $G^2$ is the transition space of the continuity of LE. That is, the Lyapunov exponent $L(\alpha, A)$ is continuous in $G^s(\mathbb S^1,\mathrm{SL}(2,\mathbb R))$ for $1<s<2$, while it has discontinuous points in $G^s(\mathbb S^1, \mathrm{SL}(

Theorems & Definitions (47)

  • Definition 1.1
  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • ...and 37 more