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Partial classification of spectrum maximizing products for pairs of $2\times2$ matrices

Piotr Laskawiec

TL;DR

The paper advances the understanding of spectrum maximizing products (SMPs) by partitioning irreducible pairs of real $2\times2$ matrices into regions $\mathcal{R}_{cross}, \mathcal{R}_{mix}, \mathcal{R}_{neg}, \mathcal{R}_{copar}$ and providing explicit SMP descriptions in each region, with a unifying ergodic-optimization perspective via Sturmian measures. It establishes algebraic and geometric criteria to identify regions, proves that SMPs in $\mathcal{R}_{cross}$ are restricted to $A$ or $B$, that mixed regions yield a small set of structured forms, and that in $\mathcal{R}_{copar}$ SMPs must be Sturmian and unique when they exist. The work also proves generic uniqueness of SMPs outside the unexplored regions and extends the results to nonnegative matrix pairs, linking combinatorial word structures (Christoffel/Sturmian) to spectral optimization. Overall, the results illuminate when SMPs are simple and Sturmian, clarify the role of ergodic optimization in JSR problems, and delineate where more complex, non-Sturmian behavior may occur, including the boundary with unexplored regions.

Abstract

Experiments suggest that typical finite sets of square matrices admit spectrum maximizing products (SMPs): that is, products that attain the joint spectral radius (JSR). Furthermore, those SMPs are often combinatorially "simple." In this paper, we consider pairs of real $2 \times 2$ matrices. We identify regions in the space of such pairs where SMPs are guaranteed to exist and to have a simple structure. We also identify another region where SMPs may fail to exist (in fact, this region includes all known counterexamples to the finiteness conjecture), but nevertheless a Sturmian maximizing measure exists. Though our results apply to a large chunk of the space of pairs of $2 \times 2$ matrices, including for instance all pairs of non-negative matrices, they leave out certain "wild" regions where more complicated behavior is possible.

Partial classification of spectrum maximizing products for pairs of $2\times2$ matrices

TL;DR

The paper advances the understanding of spectrum maximizing products (SMPs) by partitioning irreducible pairs of real matrices into regions and providing explicit SMP descriptions in each region, with a unifying ergodic-optimization perspective via Sturmian measures. It establishes algebraic and geometric criteria to identify regions, proves that SMPs in are restricted to or , that mixed regions yield a small set of structured forms, and that in SMPs must be Sturmian and unique when they exist. The work also proves generic uniqueness of SMPs outside the unexplored regions and extends the results to nonnegative matrix pairs, linking combinatorial word structures (Christoffel/Sturmian) to spectral optimization. Overall, the results illuminate when SMPs are simple and Sturmian, clarify the role of ergodic optimization in JSR problems, and delineate where more complex, non-Sturmian behavior may occur, including the boundary with unexplored regions.

Abstract

Experiments suggest that typical finite sets of square matrices admit spectrum maximizing products (SMPs): that is, products that attain the joint spectral radius (JSR). Furthermore, those SMPs are often combinatorially "simple." In this paper, we consider pairs of real matrices. We identify regions in the space of such pairs where SMPs are guaranteed to exist and to have a simple structure. We also identify another region where SMPs may fail to exist (in fact, this region includes all known counterexamples to the finiteness conjecture), but nevertheless a Sturmian maximizing measure exists. Though our results apply to a large chunk of the space of pairs of matrices, including for instance all pairs of non-negative matrices, they leave out certain "wild" regions where more complicated behavior is possible.

Paper Structure

This paper contains 13 sections, 42 theorems, 103 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Some notes on the regions
  3. Simultaneous Conjugacy classes
  4. Geometric description of a few regions of interest
  5. Algebraic description of regions $\mathcal{R}_{cross}$, $\mathcal{R}_{copar}$ and $\mathcal{S}_{anti}$
  6. What are the possible SMPs when $(A,B) \in \mathcal{R}_{cross}$?
  7. What are the SMPs if $(A,B) \in \mathcal{R}_{mix}$?
  8. What are the SMPs if $(A,B) \in \mathcal{R}_{neg}$?
  9. Ergodic Optimization and Joint Spectral Radius
  10. What are the possible SMPs if $(A,B) \in \mathcal{R}_{copar}$ ?
  11. Generic Uniqueness
  12. The unexplored regions
  13. Non-negative matrices

Key Result

theorem 1.2

berger-wang For a bounded set of matrices $\mathcal{A}$

Figures (4)

  • Figure 1: Example configurations of the translation axes.
  • Figure 2: For $n=5$, graph of $\frac{nx\log |x|}{\log c}$ (blue), the vectors (red) and tangent lines at $v_5,-v_5$ (black).
  • Figure 3: The Christoffel tree
  • Figure :

Theorems & Definitions (81)

  • definition 1.1
  • theorem 1.2
  • definition 1.3
  • definition 1.8
  • theorem 1.10
  • theorem 2.1
  • theorem 2.2
  • theorem 2.3
  • theorem 2.4
  • proof
  • ...and 71 more