Dichotomy spectrum and almost topological conjugacy on nonautonomus unbounded difference systems

Abstract

We construct a bijection between the solutions of a linear system of nonautonomous difference equations which is uniformly asymptotically stable and its unbounded perturbation. The key idea used to made this bijection is to consider the crossing times of the solutions with the unit sphere. This approach prompt us to introduce the concept of almost topological conjugacy in this nonautonomous framework. This task is carried out by simplifying both systems through a spectral approach of the notion of almost reducibility combined with suitable technical assumptions.

Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line

“…We point out that there exist other linearization results which follow ideas and methods different to the Palmer's construction. In particular, we highlight the approach based in the crossing times with the unit ball which has been employed with several variations in [5,8,19].…”

“…Step 2: Constructing H and G. By the uniqueness of solutions we have that 8) and the reader can verify that…”

Linearization of a nonautonomous unbounded system with nonuniform contraction: A spectral approach

“…If we denote g(t, y) = S −1 (δ, t)f (t, S(δ, t)y), then g ∈ A 2 and additionally Proof. This proof follows the lines of the papers [5] and [9]. We take the auxiliar function f 0 (t, x) = f (t, x) − f (t, 0) and we affirm that when f ∈ A 1 then f 0 ∈ A 2 .…”

“…We emphasize that only few results of topological conjugacy consider the unbounded nonlinearity. To the best of our knowledge, this property of unboundedness is only considered in [9], [5] (in a differential and discrete uniform context, respectively) and [17] in an impulsive framework.…”

Nonuniform almost reducibility of nonautonomous linear differential equations

“…In the continuous and discrete cases, the concept of contractibility has been applied in some results of topological equivalence and almost topological equivalence respectively (see [15], [6]). The major contribution of [14] is to prove that the contractible set of a linear system (1) is its Sacker and Sell spectrum (see [18]).…”