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Effective Gaps between singular values of non-stationary matrix products subject to non-degenerate noise

Sam Bednarski, Jonathan DeWitt, Anthony Quas

TL;DR

The paper addresses the behavior of non-stationary products of bounded matrices perturbed by small additive noise and proves that the gaps between the logarithms of consecutive singular values grow at least like $\varepsilon^2 n$ both in expectation and almost surely. It extends the Gorodetski–Kleptsyn entropy framework to a fibered setting on miniflags, introducing fibered and pointwise entropy tools and a Fubini-type argument to relate volume distortion to singular-value gaps. A key contribution is a quantitative uniform entropy gap of order $\varepsilon^2/M^2$ for noise on miniflags, which yields universal lower bounds on all consecutive singular-value gaps in the random product. The results provide a robust, versatile approach for studying ergodic properties of perturbations of dynamical systems and may have broader applications in related problems involving non-stationary random linear cocycles.

Abstract

We study the singular values and Lyapunov exponents of non-stationary random matrix products subject to small, absolutely continuous, additive noise. Consider a fixed sequence of matrices of bounded norm. Independently perturb the matrices by additive noise distributed according to Lebesgue measure on matrices with norm less than $ε$. Then the gaps between the logarithms of the singular values of the random product of $n$ of these matrices are all of order at least $ε^2n$, both in expectation; and almost surely for large $n$. To prove this, we develop recent work of Gorodetski and Kleptsyn \cite{gorodetski2023nonstationary}. That paper gives a very flexible method, based on relative entropy, for showing that a non-stationary product of matrices in SL(d,R) has a strictly positive Lyapunov exponent. We extend their work in two ways, firstly by making the estimates quantitative in the context of absolutely continuous distributions, giving the universal estimates described above; and secondly by developing a fibered version of their methods, working on flag bundles instead of the projective space to estimate gaps between arbitrary consecutive exponents. Our methods retain much of the flexibility of those of Gorodetski and Kleptsyn, and we hope that they will find application in other related problems.

Effective Gaps between singular values of non-stationary matrix products subject to non-degenerate noise

TL;DR

The paper addresses the behavior of non-stationary products of bounded matrices perturbed by small additive noise and proves that the gaps between the logarithms of consecutive singular values grow at least like both in expectation and almost surely. It extends the Gorodetski–Kleptsyn entropy framework to a fibered setting on miniflags, introducing fibered and pointwise entropy tools and a Fubini-type argument to relate volume distortion to singular-value gaps. A key contribution is a quantitative uniform entropy gap of order for noise on miniflags, which yields universal lower bounds on all consecutive singular-value gaps in the random product. The results provide a robust, versatile approach for studying ergodic properties of perturbations of dynamical systems and may have broader applications in related problems involving non-stationary random linear cocycles.

Abstract

We study the singular values and Lyapunov exponents of non-stationary random matrix products subject to small, absolutely continuous, additive noise. Consider a fixed sequence of matrices of bounded norm. Independently perturb the matrices by additive noise distributed according to Lebesgue measure on matrices with norm less than . Then the gaps between the logarithms of the singular values of the random product of of these matrices are all of order at least , both in expectation; and almost surely for large . To prove this, we develop recent work of Gorodetski and Kleptsyn \cite{gorodetski2023nonstationary}. That paper gives a very flexible method, based on relative entropy, for showing that a non-stationary product of matrices in SL(d,R) has a strictly positive Lyapunov exponent. We extend their work in two ways, firstly by making the estimates quantitative in the context of absolutely continuous distributions, giving the universal estimates described above; and secondly by developing a fibered version of their methods, working on flag bundles instead of the projective space to estimate gaps between arbitrary consecutive exponents. Our methods retain much of the flexibility of those of Gorodetski and Kleptsyn, and we hope that they will find application in other related problems.

Paper Structure

This paper contains 21 sections, 4 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Related Results
  3. Outline of the Approach
  4. Preliminaries
  5. Miniflags
  6. Basic Facts Concerning Entropy
  7. The relationship between entropy and volume distortion
  8. Miniflag volume distortion and singular values
  9. Entropy of Random Bundle Maps
  10. Tools for Bounding Entropy Below
  11. Pointwise Entropy
  12. The Fubini Argument
  13. Mean miniflag volume distortion implies singular value gaps
  14. Mean miniflag fiber distortion implies mean singular value gaps
  15. Entropy of the induced random bundle map on miniflags
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\mu$ be an absolutely continuous probability measure on $M_{d\times d}(\mathbb{R})$, the space of $d$ by $d$ matrices, with continuous density $\phi$ such that for some $C>0$, $\phi(A)\le C/\|A\|^{d^2+1}$ for all $A\in M_{d\times d}(\mathbb{R})$. There exists $C_{\phi}$ with the following prope

Theorems & Definitions (40)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.3
  • Definition 2.4
  • proof
  • proof
  • proof
  • proof
  • proof
  • Definition 3.1
  • ...and 30 more