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Lyapunov Exponents for Sparsely Coupled Linear Cocycles

Reza Rastegar

TL;DR

This work develops a toolkit for bounding and, in favorable cases, computing the top Lyapunov exponent $\gamma_1$ for products of real matrices with sparse or block-structured zero patterns. Central ideas combine diagonal-block growth with a generalized Furstenberg–Kifer reduction and a novel shape-graph formalism to capture sparsity, yielding explicit bounds $\gamma_1(\mathcal{A}) \le \beta + \log k$ (or $\beta + \log k_*$) that separate genuine diagonal growth from combinatorial proliferation. The results apply to tempered and stationary ergodic cocycles and extend to matrix perturbations, including rank-one and low-rank updates, where perturbations induce quotient cocycles that determine the remaining exponents. The shape-graph perspective provides a practical, computable framework for analyzing sparsely coupled linear systems and offers clear directions for refinements beyond acyclic structures and for handling overlaps or cancellations, with implications for transfer-matrix models and perturbation theory in control and dynamical systems.

Abstract

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.

Lyapunov Exponents for Sparsely Coupled Linear Cocycles

TL;DR

This work develops a toolkit for bounding and, in favorable cases, computing the top Lyapunov exponent for products of real matrices with sparse or block-structured zero patterns. Central ideas combine diagonal-block growth with a generalized Furstenberg–Kifer reduction and a novel shape-graph formalism to capture sparsity, yielding explicit bounds (or ) that separate genuine diagonal growth from combinatorial proliferation. The results apply to tempered and stationary ergodic cocycles and extend to matrix perturbations, including rank-one and low-rank updates, where perturbations induce quotient cocycles that determine the remaining exponents. The shape-graph perspective provides a practical, computable framework for analyzing sparsely coupled linear systems and offers clear directions for refinements beyond acyclic structures and for handling overlaps or cancellations, with implications for transfer-matrix models and perturbation theory in control and dynamical systems.

Abstract

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 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.
Paper Structure (12 sections, 12 theorems, 126 equations, 2 figures)

This paper contains 12 sections, 12 theorems, 126 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Block-triangular matrices
  3. Application to matrix perturbations
  4. Shape graphs
  5. Theorem \ref{['th-shapes']} in practice
  6. Proof of Theorem \ref{['th-shapes']}
  7. Discussion and outlook
  8. Energy--entropy structure and regimes.
  9. Possible refinements of the entropy term.
  10. Beyond acyclic shape graphs.
  11. Overlapping supports and cancellations.
  12. Practical use.

Key Result

Proposition 1.1

pinkus Let ${\mathcal{A}}=(A_k)_{k\in{\mathbb N}}\in{\mathcal{M}}_d^{\mathbb N}$ be a random, stationary, and ergodic sequence of $d\times d$ invertible matrices with $A_k(i,j) = 0$ for $i > j$ and such that f_cond holds. Then, where $[d]:=\{1,\ldots,d\}.$

Figures (2)

  • Figure 1: The shape graph for \ref{['exm1-d1']} and \ref{['exm1-v1']}.
  • Figure 2: Two different graphs for the matrices described by \ref{['two_exm']}

Theorems & Definitions (29)

  • Proposition 1.1
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Remark 2.5
  • proof : Proof of Proposition \ref{['thm101']}
  • Proposition 3.1
  • Remark 3.2: Varying right vector: two extendable regimes
  • ...and 19 more