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arxiv: 2312.03181 · v2 · submitted 2023-12-05 · 🧮 math.DS

Universal Gap Growth for Lyapunov Exponents of Perturbed Matrix Products

Jason Atnip , Gary Froyland , Cecilia Gonz\'alez-Tokman , Anthony Quas This is my paper

Pith reviewed 2026-05-24 05:12 UTC · model grok-4.3

classification 🧮 math.DS
keywords Lyapunov exponentsmatrix cocyclesadditive perturbationsgap boundsrandom dynamical systemssequential dynamical systemsOseledets spectrum
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The pith

Additive random perturbations of matrix cocycles force positive lower bounds on gaps between consecutive Lyapunov exponents that depend only on perturbation size.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies bounded sequences of d by d matrices under small additive random perturbations and shows that the Lyapunov exponents of the resulting perturbed cocycle separate. In dimensions two and three the separation is bounded below by explicit positive quantities that depend solely on the size of the perturbation. In all dimensions the existence of some positive universal lower bound is proved. Both results hold uniformly for every choice of the original matrix sequence and require no stationarity or other regularity on that sequence, so they apply equally to random and to sequential dynamical systems.

Core claim

For any dimension d and any bounded sequence of d by d matrices, an additive random perturbation produces a new cocycle whose Lyapunov exponents satisfy a uniform positive lower bound on the gaps between consecutive values. In dimensions 2 and 3 the bound is explicit and depends only on the scale of the perturbation; in higher dimensions existence of a positive bound independent of the original sequence is shown. The bounds require only that the perturbation be random with positive probability of sufficient directional spread and make no stationarity assumptions on the unperturbed sequence.

What carries the argument

The additive random perturbation of the matrix cocycle, which with positive probability spreads directions enough to separate the Lyapunov exponents uniformly over all original sequences.

Load-bearing premise

The perturbations are additive and random with positive probability of sufficient directional spread.

What would settle it

A fixed perturbation size and distribution together with some bounded matrix sequence for which the Lyapunov exponents of the perturbed cocycle remain arbitrarily close.

read the original abstract

We study the quantitative simplicity of the Lyapunov spectrum of $d$-dimensional bounded matrix cocycles subjected to additive random perturbations. In dimensions 2 and 3, we establish explicit lower bounds on the gaps between consecutive Lyapunov exponents of the perturbed cocycle, depending only on the scale of the perturbation. In arbitrary dimensions, we show existence of a universal lower bound on these gaps. A novelty of this work is that the bounds provided are uniform over all choices of the original sequence of matrices. Furthermore, we make no stationarity assumptions on this sequence. Hence, our results apply to random and sequential dynamical systems alike.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript proves lower bounds on gaps between consecutive Lyapunov exponents for d-dimensional bounded matrix cocycles under additive random perturbations. In dimensions 2 and 3, explicit lower bounds are obtained that depend only on the perturbation scale; in arbitrary dimensions, the existence of a universal (positive) lower bound is established. Both results are uniform over all choices of the original bounded matrix sequence and impose no stationarity assumptions on the sequence.

Significance. If the proofs hold, the uniformity over arbitrary (non-stationary) sequences constitutes a genuine strengthening of existing results on Lyapunov spectrum simplicity under perturbation. The explicit constants in low dimensions and the dimension-independent existence statement would be of direct use in both random and sequential dynamical systems.

minor comments (3)
  1. [§2.2] §2.2: the definition of the perturbed cocycle (equation (2.3)) should explicitly record the support condition on the random perturbation that is used in the gap estimates.
  2. [Theorem 1.1] Theorem 1.1: the dependence of the constant C_d on dimension d is not quantified; a remark on whether the bound deteriorates with d would clarify the scope of the higher-dimensional result.
  3. [Figure 1] Figure 1: the caption does not indicate the dimension or the perturbation scale used in the numerical illustration.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance as a strengthening of existing results on Lyapunov spectrum simplicity, and recommendation for minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents a mathematical proof establishing explicit and universal lower bounds on gaps between Lyapunov exponents for perturbed matrix cocycles, uniform over arbitrary bounded sequences without stationarity assumptions. The abstract and available claims describe direct analytic results derived from the perturbation structure and dimension-specific arguments, with no indication of self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the central claims back to the inputs by construction. The derivation appears self-contained as a rigorous existence and bound proof under the stated random additive perturbation hypothesis.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

This is a pure mathematical theorem in ergodic theory and perturbation analysis. No free parameters are introduced. The result rests on standard domain assumptions about bounded linear cocycles and additive random perturbations.

axioms (2)
  • domain assumption The original sequence consists of bounded matrices
    Explicitly stated in the abstract as the setting for the cocycles.
  • domain assumption Perturbations are additive and random
    Central modeling choice that enables the gap creation; invoked throughout the claimed results.

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