[Paper Review] Extremal ergodic measures and the finiteness property of matrix semigroups

Xiongping Dai, Yu Huang+1 more|arXiv (Cornell University)|Jul 1, 2011

Mathematical Dynamics and Fractals20 references3 citations

TL;DR

This paper establishes a necessary and sufficient condition for the spectral finiteness property of a finite set of complex matrices: the joint spectral radius equals the spectral radius of some finite product of matrices if and only if there exists an extremal ergodic measure for the matrix semigroup that has a periodic density point. The proof uses extremal norms and the Kingman subadditive ergodic theorem to link ergodic theory with the finiteness property, revealing that periodicity in extremal measures characterizes when the joint spectral radius is achieved by a finite product.

ABSTRACT

Let $\bS=\{S_1,...,S_K\}$ be a finite set of complex $d imes d$ matrices and $\varSigma_{K}^+$ the compact space of all one-sided infinite sequences $i_{\bcdot}\colon\mathbb{N} ightarrow\{1,...,K\}$. An ergodic probability $μ_*$ of the Markov shift $θ\colon\varSigma_{K}^+ ightarrow\varSigma_{K}^+;\ i_{\bcdot}\mapsto i_{\bcdot+1}$, is called "extremal" for $\bS$, if $ρ(\bS)=\lim_{n o\infty}\sqrt[n]{ orm{S_{i_1}...S_{i_n}}}$ holds for $μ_*$-a.e. $i_{\bcdot}\in\varSigma_{K}^+$, where $ρ(\bS)$ denotes the generalized/joint spectral radius of $\bS$. Using extremal norm and Kingman subadditive ergodic theorem, it is shown that $\bS$ has the spectral finiteness property (i.e. $ρ(\bS)=\sqrt[n]{ρ(S_{i_1}...S_{i_n})}$ for some finite-length word $(i_1,...,i_n)$) if and only if for some extremal measure $μ_*$ of $\bS$, it has at least one periodic density point $i_{\bcdot}\in\varSigma_{K}^+$.

Motivation & Objective

• To determine the precise condition under which the joint spectral radius of a finite matrix family is achieved by a finite product of matrices (the finiteness property).
• To characterize the spectral finiteness property using ergodic theory and the structure of invariant measures on symbolic dynamics.
• To clarify the role of extremal ergodic measures—particularly their periodic density points—in determining whether the generalized spectral radius is computable via finite products.
• To provide a theoretical foundation for algorithms that compute the joint spectral radius by identifying when the finiteness property holds.

Proposed method

• Define an extremal ergodic measure μ* as one for which the joint spectral radius ρ(S) equals the almost-sure growth rate of ||Si₁⋯Sin|| under μ*.
• Apply the Kingman subadditive ergodic theorem to establish the existence of such extremal measures for any finite matrix family S.
• Use extremal norms to relate the spectral radius to matrix product growth and to analyze the behavior of products along typical sequences.
• Prove that the finiteness property holds if and only if some extremal measure μ* has a periodic point in its support (i.e., a periodic sequence i· with i_{n+π} = i_n for all n).
• Leverage the Gel’fand formula and properties of cylinder sets to derive a contradiction if no such periodic point exists, thereby proving necessity.
• Apply the result to canonical Markovian measures μ_p,P to derive sufficient conditions for the finiteness property based on ergodicity and support structure.

Experimental results

Research questions

• RQ1 Under what conditions does the joint spectral radius of a finite set of matrices equal the spectral radius of some finite product of those matrices?
• RQ2 How is the existence of periodic sequences in the support of extremal ergodic measures related to the finiteness property of matrix semigroups?
• RQ3 Can the finiteness property be characterized purely through the topological and dynamical structure of extremal invariant measures on the symbolic space Σ_K^+?
• RQ4 What role do non-observable, non-Markovian extremal measures play in the failure of the finiteness property?
• RQ5 How does the presence of periodic density points in extremal measures affect the stability and computability of the joint spectral radius?

Key findings

• The spectral finiteness property holds for a matrix family S if and only if there exists an extremal ergodic measure μ* for S that has at least one periodic density point in its support.
• If an extremal measure μ* has a periodic density point i· with period π, then the joint spectral radius satisfies ρ(S) = √[π]{ρ(S_{i₁'}⋯S_{i_π'})}, proving the finiteness property.
• The existence of such a periodic point implies that the growth rate of matrix products along the periodic sequence achieves the joint spectral radius.
• For canonical Markovian measures μ_p,P, if such a measure is extremal, then the finiteness property holds, providing a sufficient condition based on stochastic dynamics.
• The failure of the finiteness property is linked to the absence of periodic points in all extremal ergodic measures, implying that non-observable, non-periodic measures must be considered in counterexamples.
• The result shows that the topological structure of extremal measures—specifically periodicity in their support—is a decisive factor in determining whether the joint spectral radius is computable via finite products.