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The joint spectral radius is pointwise Hölder continuous

Jeremias Epperlein, Fabian Wirth

TL;DR

This work studies the regularity of the joint spectral radius $\rho(\mathcal{M})$ as a function on the space of compact matrix sets $\mathcal{H}(d)$. By combining ε-inflation analysis, flag-based norm constructions for reducible systems, and perturbation methods, it establishes pointwise Hölder continuity with exponent $\frac{1}{d^2+d}$ and, for finite sets, exponents arbitrarily close to $\frac{1}{d}$; in dimension two, it proves local Hölder continuity with exponent $\frac{1}{6}$ on $\mathcal{H}_{>0}(2)$. The continuous-time analogue yields Hölder regularity for the maximal Lyapunov exponent with related exponents. Overall, the paper advances the understanding of how perturbations of matrix-sets affect growth rates, with implications for numerical analysis and dynamical-systems applications. $\,$

Abstract

We show that the joint spectral radius is pointwise Hölder continuous. In addition, the joint spectral radius is locally Hölder continuous for $\varepsilon$-inflations. In the two-dimensional case, local Hölder continuity holds on the matrix sets with positive joint spectral radius.

The joint spectral radius is pointwise Hölder continuous

TL;DR

This work studies the regularity of the joint spectral radius as a function on the space of compact matrix sets . By combining ε-inflation analysis, flag-based norm constructions for reducible systems, and perturbation methods, it establishes pointwise Hölder continuity with exponent and, for finite sets, exponents arbitrarily close to ; in dimension two, it proves local Hölder continuity with exponent on . The continuous-time analogue yields Hölder regularity for the maximal Lyapunov exponent with related exponents. Overall, the paper advances the understanding of how perturbations of matrix-sets affect growth rates, with implications for numerical analysis and dynamical-systems applications.

Abstract

We show that the joint spectral radius is pointwise Hölder continuous. In addition, the joint spectral radius is locally Hölder continuous for -inflations. In the two-dimensional case, local Hölder continuity holds on the matrix sets with positive joint spectral radius.
Paper Structure (12 sections, 25 theorems, 134 equations, 1 figure)

This paper contains 12 sections, 25 theorems, 134 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Preliminaries
  4. Elementary inequalities
  5. Hölder continuity
  6. Reducibility and norms
  7. $\varepsilon$-inflation
  8. Pointwise Hölder continuity
  9. Local $\frac{1}{d^2+d}$-Hölder-continuity of the JSR in dimension two
  10. Continuous-Time
  11. Conclusion
  12. Proof of \ref{['prop:specrad-lowerbound']}

Key Result

Theorem 1

For $A,B \in \mathbb{C}^{d \times d}$ we have

Figures (1)

  • Figure 1: Summary of properties proved in the paper versus those conjectured by the authors. In the left of a box the proved results are stated, whereas in the right the respective conjecture is shown. The statements labeling the implication arrows between boxes concern statements that show that the lower level statements (results as well as conjectures) imply the upper level.

Theorems & Definitions (49)

  • Theorem 1: Elsner
  • Proposition 2
  • Conjecture 3
  • Proposition 4
  • proof
  • Lemma 5
  • Remark 6
  • Example 7
  • Definition 8
  • Lemma 9
  • ...and 39 more