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Computation of Lyapunov exponents of matrix products

Aihua Fan, Evgeny Verbitskiy

TL;DR

The paper identifies a class of matrix-product Lyapunov exponents with a rank-one anchor $A_0=\mathbf{u}\mathbf{v}'$ for which an explicit closed form is possible under genericity and nondegeneracy conditions. By decomposing the sequence $\omega$ into return words to $0$ and introducing exact return-word frequencies $F_{\mathbf w}$ and the base frequency $\rho_0$, the authors derive a universal formula: $L(\omega)=\rho_0\log|\mathbf{v}'\mathbf{u}|+\sum_{\mathbf w\in\mathcal{R}} F_{\mathbf w}\log\frac{|\mathbf{v}'A_{\mathbf w}\mathbf{u}|}{|\mathbf{v}'\mathbf{u}|}$. They show this framework applies to deterministic nonnegative matrices, random products, and, notably, matrices selected by primitive substitutive sequences and by $\mathcal{B}$-free integers, with concrete examples (Fibonacci, Thue–Morse, Tribonacci) yielding explicit exponents. The work also develops methods to compute exact frequencies and extends to multifractal analysis of weighted Birkhoff averages, including Fibonacci and Möbius-type weights, offering practical benchmarks and analytic clarity for Lyapunov-analytic computations in structured symbolic dynamics.

Abstract

For $m$ given square matrices $A_0, A_1, \cdots, A_{m-1}$ ($m\ge 2$), one of which is assumed to be of rank $1$, and for a given sequence $(ω_n)$ in $\{0,1, \cdots, m-1\}^\mathbb{N}$, the following limit, if it exists, $$L(ω):=\lim_{n\to \infty} \frac 1n \log \|A_{ω_0} A_{ω_2}\cdots A_{ω_{n-1}}\|$$ defines the Lyapunov exponent of the sequence of matrices $(A_{ω_n})_{n\ge 0}$. It is proved that the Lyapunov exponent $L(ω)$ has a closed-form expression under certain conditions. One special case arises when $A_j$'s are non-negative and $ω$ is generic with respect to some shift-invariant measure; a second special case occurs when $A_j$'s (for $1\le j<m$) are invertible and $ω$ is a typical point with respect to some shift-ergodic measure. Substitutive sequences and characteristic sequences of $\mathcal{B}$-free integers are considered as examples. An application is presented for the computation of multifractal spectrum of weighted Birkhoff averages.

Computation of Lyapunov exponents of matrix products

TL;DR

The paper identifies a class of matrix-product Lyapunov exponents with a rank-one anchor for which an explicit closed form is possible under genericity and nondegeneracy conditions. By decomposing the sequence into return words to and introducing exact return-word frequencies and the base frequency , the authors derive a universal formula: . They show this framework applies to deterministic nonnegative matrices, random products, and, notably, matrices selected by primitive substitutive sequences and by -free integers, with concrete examples (Fibonacci, Thue–Morse, Tribonacci) yielding explicit exponents. The work also develops methods to compute exact frequencies and extends to multifractal analysis of weighted Birkhoff averages, including Fibonacci and Möbius-type weights, offering practical benchmarks and analytic clarity for Lyapunov-analytic computations in structured symbolic dynamics.

Abstract

For given square matrices (), one of which is assumed to be of rank , and for a given sequence in , the following limit, if it exists, defines the Lyapunov exponent of the sequence of matrices . It is proved that the Lyapunov exponent has a closed-form expression under certain conditions. One special case arises when 's are non-negative and is generic with respect to some shift-invariant measure; a second special case occurs when 's (for ) are invertible and is a typical point with respect to some shift-ergodic measure. Substitutive sequences and characteristic sequences of -free integers are considered as examples. An application is presented for the computation of multifractal spectrum of weighted Birkhoff averages.
Paper Structure (28 sections, 16 theorems, 254 equations, 3 figures)

This paper contains 28 sections, 16 theorems, 254 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A general formula
  3. Decomposition of omega into return words and return word statistics.
  4. Main result
  5. A counterexample
  6. Convergence of the series $\sum_{\mathbf{w}\in \mathcal{R}}F_\mathbf{w} \log \left|\frac{\mathbf{v}'A_{\mathbf{w}} \mathbf{u} }{\mathbf{v}' \mathbf{u}}\right|$
  7. Proof of Theorem \ref{['thm:rank1']}
  8. Product of non-negative matrices: Proof of Theorem \ref{['thm:positive']}
  9. Structure of generic sequence
  10. Proof of Theorem \ref{['thm:positive']}
  11. Product of random matrices: Proof of Theorem \ref{['thm:random']}
  12. Computation of exact frequencies
  13. Computation of $F_\mathbf{w}$ by $\rho_{\mathbf{w}'}$ with $|\mathbf{w}'|\le |\mathbf{w}|$
  14. Proof of Theorem \ref{['thm:ExactF']}
  15. Examples
  16. ...and 13 more sections

Key Result

Theorem 1.1

Suppose that then the limit eq:DefLyap defining $L(\omega)$ exists. Moreover, $L(\omega)=-\infty$ if $A_{{\mathbf w}_0}\mathbf{u}=0$ or $\mathbf{v}'A_{\mathbf{w}} \mathbf{u}=0$ for some $\mathbf{w}\in\mathcal{R}$; otherwise, $L(\omega)$ is given by

Figures (3)

  • Figure 1: Graphs of $\psi(\beta)$ and multifractal spectrum in the case of $f(x_0, x_1) = x_0x_1$ with Fibonacci weight.
  • Figure 2: Graphs of $\psi(\beta)$ and $\dim \{x: \Phi(x)=\alpha\}$ in the case: $f(x_0, x_1) = x_0x_1$ and $w_n=\mu(n)^2$.
  • Figure 3: Graph of $\psi'(\beta)$ in the case: $f(x_0, x_1) = x_0x_1$ and $w_n=\mu(n)^2$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • ...and 11 more