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Discrete $μ$-dichotomy spectrum: beyond uniformity and new insights

Álvaro Castañeda, Claudio A. Gallegos, Néstor Jara

TL;DR

This work develops a spectral theory for nonautonomous linear difference equations under $\mu$-dichotomies, introducing uniform, nonuniform, and slow nonuniform notions and two key properties (USP and UPP). It defines an intermediate spectrum $\Sigma_{\mathrm{sN}\mu\mathrm{D}}^{\mathrm{UPP}}(A)$ and posits the USPP conjecture, showing how weighted systems and optimal ratio maps yield spectral decompositions and potential invariance results. A central contribution is the spectral theorem for $\mathrm{N}\mu\mathrm{D}$ and the construction of the intermediate spectrum between slow nonuniform and nonuniform cases, along with a counterexample demonstrating that the nonuniform $\mu$-dichotomy spectrum is not preserved under weak kinematic similarity in discrete time. The findings clarify the limitations of invariance assumptions in nonautonomous spectral theory and provide tools (weighted systems, USP/UPP, and optimal ratio maps) to inform future linearization, normal-form analyses, and reducibility considerations in both discrete and continuous dynamics.

Abstract

We develop spectral theorems for nonautonomous linear difference systems, considering different types of $μ$-dichotomies, both uniform and nonuniform. In the nonuniform case, intriguing scenarios emerge -- that have been employed but whose consequences have not been thoroughly explored -- which surprisingly exhibit unconventional behavior. These particular cases motivate us to introduce two novel properties of nonautonomous systems (even in the continuous-time framework), which appear to have been overlooked in the existing literature. Additionally, we introduce a new conceptualization of a nonuniform $μ$-dichotomy spectrum, which lies between the traditional nonuniform $μ$-dichotomy spectrum and the slow nonuniform $μ$-dichotomy spectrum. Moreover, and this is particularly noteworthy, we propose a conjecture that enables the derivation of spectral theorems in this new setting. Finally, contrary to what has been believed in recent years, through the lens of optimal ratio maps, we show that the nonuniform exponential dichotomy spectrum is not preserved between systems that are weakly kinematically similar.

Discrete $μ$-dichotomy spectrum: beyond uniformity and new insights

TL;DR

This work develops a spectral theory for nonautonomous linear difference equations under -dichotomies, introducing uniform, nonuniform, and slow nonuniform notions and two key properties (USP and UPP). It defines an intermediate spectrum and posits the USPP conjecture, showing how weighted systems and optimal ratio maps yield spectral decompositions and potential invariance results. A central contribution is the spectral theorem for and the construction of the intermediate spectrum between slow nonuniform and nonuniform cases, along with a counterexample demonstrating that the nonuniform -dichotomy spectrum is not preserved under weak kinematic similarity in discrete time. The findings clarify the limitations of invariance assumptions in nonautonomous spectral theory and provide tools (weighted systems, USP/UPP, and optimal ratio maps) to inform future linearization, normal-form analyses, and reducibility considerations in both discrete and continuous dynamics.

Abstract

We develop spectral theorems for nonautonomous linear difference systems, considering different types of -dichotomies, both uniform and nonuniform. In the nonuniform case, intriguing scenarios emerge -- that have been employed but whose consequences have not been thoroughly explored -- which surprisingly exhibit unconventional behavior. These particular cases motivate us to introduce two novel properties of nonautonomous systems (even in the continuous-time framework), which appear to have been overlooked in the existing literature. Additionally, we introduce a new conceptualization of a nonuniform -dichotomy spectrum, which lies between the traditional nonuniform -dichotomy spectrum and the slow nonuniform -dichotomy spectrum. Moreover, and this is particularly noteworthy, we propose a conjecture that enables the derivation of spectral theorems in this new setting. Finally, contrary to what has been believed in recent years, through the lens of optimal ratio maps, we show that the nonuniform exponential dichotomy spectrum is not preserved between systems that are weakly kinematically similar.
Paper Structure (13 sections, 20 theorems, 114 equations, 1 figure)

This paper contains 13 sections, 20 theorems, 114 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Novelty and Main Results
  3. Notions of Dichotomy and Properties
  4. Properties of Each Notion of Dichotomy
  5. USPP Conjecture
  6. Spectrum and Spectral Theorems
  7. The Slow Nonuniform Case
  8. Consequences of USPP Conjecture
  9. Optimal Dichotomy Constants
  10. Kinematic Similarity and Spectra Non-invariance
  11. Continuous Nonuniform Kinematic Similarity
  12. Discrete Nonuniform Kinematic Similarity
  13. The Non-invariance Problem

Key Result

Lemma 2.6

Assume that system 700 has a dichotomy (either ${\mu\mathrm{D}}$, ${\mathrm{N}\mu\mathrm{D}}$ or ${\mathrm{sN}\mu\mathrm{D}}$) with invariant projector $n\mapsto\mathrm{P}(n)$. Then, $\mathrm{im}\, \mathrm{P}\subset \mathcal{S}$ and $\ker \mathrm{P}\subset \mathcal{U}$.

Figures (1)

  • Figure 1: Sketch of the graph of the functions $\mathfrak{un}_{\mathrm{P}}$ and $\mathfrak{st}_{\mathrm{P}}$ defined on the spectral gaps $(-\infty,a_1)$, $(b_1,a_2)$, and $(b_2,+\infty)$.

Theorems & Definitions (77)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • Remark 2.8
  • Lemma 2.9
  • ...and 67 more