The new notion of Bohl dichotomy for non-autonomous difference equations and its relation to exponential dichotomy

Abstract

In A. Czornik et al. [Spectra based on Bohl exponents and Bohl dichotomy for non-autonomous difference equations, J. Dynam. Differ. Equ. (2023)] the concept of Bohl dichotomy is introduced which is a notion of hyperbolicity for linear non-autonomous difference equations that is weaker than the classical concept of exponential dichotomy. In the class of systems with bounded invertible coefficient matrices which have bounded inverses, we study the relation between the set $\mathrm{BD}$ of systems with Bohl dichotomy and the set $\mathrm{ED}$ of systems with exponential dichotomy. It can be easily seen from the definition of Bohl dichotomy that $\mathrm{ED}\subseteq \mathrm{BD}$. Using a counterexample we show that the closure of $\mathrm{ED}$ is not contained in $\mathrm{BD}$. The main result of this paper is the characterization $\mathrm{int}\mathrm{BD}=\mathrm{ED}$. The proof uses upper triangular normal forms of systems which are dynamically equivalent and utilizes a diagonal argument to choose subsequences of perturbations each of which is constructed with the Millionshikov Rotation Method. An Appendix describes the Millionshikov Rotation Method in the context of non-autonomous difference equations as a universal tool.

KEYWORDS:

• Millionshikov rotation method
• non-autonomous difference equations
• hyperbolicity
• Bohl dichotomy
• exponential dichotomy

AMS SUBJECT CLASSIFICATIONS:

• 37D25
• 39A06

Disclosure statement

No potential conflict of interest was reported by the author(s).