Lyapunov Exponents for Sparsely Coupled Linear Cocycles

This paper studies structured products of real matrices for which the top Lyapunov exponent can be accessed by reducing the dynamics to an amenable generalization of upper triangular matrices. Exploiting prescribed zero patterns (including block-triangularity and sparse decompositions, conveniently encoded by a directed sparsity graph), we obtain explicit, computable bounds and, in favorable cases, formulas for $γ_1$ by combining deterministic triangular controls with a suitable refinement of the Furstenberg–Kifer lemma for block-triangular products. The estimates apply both to tempered (possibly deterministic) sequences and to stationary ergodic random cocycles under standard integrability. We also discuss applications to perturbation models for linear systems, including low-rank updates, where the reduction converts the problem to lower-dimensional or scalar cocycles.

💡 Research Summary

The paper investigates the top Lyapunov exponent γ₁ of products of real d × d matrices under structural constraints that make the problem tractable. The authors focus on two main types of structure: (i) upper‑triangular (or more generally block‑upper‑triangular) matrices, and (ii) sparsity patterns encoded by a directed graph. Both deterministic “tempered” sequences and stationary ergodic random cocycles satisfying the usual integrability condition are considered.

The first technical contribution (Proposition 2.1 and Corollary 2.2) provides two‑sided deterministic bounds for γ₁ when the matrices are upper‑triangular. For each diagonal entry i they define the lim sup and lim inf growth rates α⁺ᵢ and α⁻ᵢ of the scalar cocycle formed by the product of the (i,i) entries. They then prove
 maxᵢ α⁻ᵢ ≤ γ₁ ≤ maxᵢ