The new notion of Bohl dichotomy for nonautonomous difference equations and its relation to exponential dichotomy

Abstract

Bohl dichotomy is a notion of hyperbolicity for linear nonautonomous 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 $\operatorname{int}\mathrm{BD} = \mathrm{ED}$. 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 nonautonomous difference equations as a universal tool.