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Hyperbolicity for one-frequency analytic quasi-periodic (Hermitian)-symplectic cocycles

Duxiao Wang, Disheng Xu, Qi Zhou

TL;DR

This work advances the understanding of hyperbolicity for real analytic one-frequency quasi-periodic cocycles valued in ${\rm Sp}(4,\mathbb{R})$ and Hermitian-symplectic groups by proving the existence of an open dense set where the Lyapunov spectrum is simple or accelerations are nonzero, partially answering Avila's open problem. The authors develop a suite of analytic tools, including block-diagonalization in both symplectic and Hermitian-symplectic settings and an Analytic Sylvester Inertia theorem, to reduce to tractable $SL(2,\mathbb{R})$-blocks and perturb them to achieve hyperbolicity. They show that simple Lyapunov spectra are open and dense within targeted regularity classes and that uniformly hyperbolic cocycles are dense in the regular set, providing a robust pathway to hyperbolicity via analytic perturbations. The results have implications for spectral theory and dynamical applications where quasi-periodic cocycles arise, offering new avenues to transfer hyperbolicity and spectral rigidity from base dynamics to higher-dimensional structure groups.

Abstract

We demonstrate the existence of an open dense subset within the class of real analytic one-frequency quasi-periodic $\mathrm{\Sp}(4,\mathbb{R})$-cocycles, characterized by either the distinctness of all their Lyapunov exponents or the non-zero nature of all their accelerations, which partially answers an open problem raised by A. Avila.

Hyperbolicity for one-frequency analytic quasi-periodic (Hermitian)-symplectic cocycles

TL;DR

This work advances the understanding of hyperbolicity for real analytic one-frequency quasi-periodic cocycles valued in and Hermitian-symplectic groups by proving the existence of an open dense set where the Lyapunov spectrum is simple or accelerations are nonzero, partially answering Avila's open problem. The authors develop a suite of analytic tools, including block-diagonalization in both symplectic and Hermitian-symplectic settings and an Analytic Sylvester Inertia theorem, to reduce to tractable -blocks and perturb them to achieve hyperbolicity. They show that simple Lyapunov spectra are open and dense within targeted regularity classes and that uniformly hyperbolic cocycles are dense in the regular set, providing a robust pathway to hyperbolicity via analytic perturbations. The results have implications for spectral theory and dynamical applications where quasi-periodic cocycles arise, offering new avenues to transfer hyperbolicity and spectral rigidity from base dynamics to higher-dimensional structure groups.

Abstract

We demonstrate the existence of an open dense subset within the class of real analytic one-frequency quasi-periodic -cocycles, characterized by either the distinctness of all their Lyapunov exponents or the non-zero nature of all their accelerations, which partially answers an open problem raised by A. Avila.

Paper Structure

This paper contains 28 sections, 40 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Analytic diagonalization of Hermitian matrix and Analytic Sylvester Inertia Theorem
  4. Difficulty, ideas of the proof
  5. Analytic block-diagonalization of partially hyperbolic cocycle.
  6. Distinctness of the Lyapunov exponents.
  7. Preliminaries
  8. Continued fraction expansion
  9. Fibered rotation number
  10. Uniform hyperbolicity and dominated splitting
  11. Analytic Block diagonalization
  12. Proof of Theorem \ref{['UH DENSITY']}
  13. Analytic Block diagonalization in symplectic groups
  14. Holomorphic structure of $Gr(n,{\mathbb C}^d)$
  15. Real-analytic structure of $Gr(n,{\mathbb R}^d)$
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $\alpha\in {\mathbb R}\backslash{\mathbb Q}$. The cocycles $(\alpha,A)$ with simple Lyapunov exponents are open and dense in the set of $1$-regular or $2$-regular $\mathrm{{Sp}}(4,\mathbb{R})$-cocycles, i.e., in the set

Theorems & Definitions (78)

  • Example 1: Derivative cocycles
  • Example 2: Random products of matrices (random walks) on a matrix Lie group
  • Example 3: Quasiperiodic cocycles
  • Theorem 1.1
  • Corollary 1.1
  • Remark 1.1
  • Theorem 1.2
  • Proposition 1.1
  • Theorem 1.3
  • Definition 1.1
  • ...and 68 more