Fine Structure of the Dichotomy Spectrum

Abstract

The dichotomy spectrum is a crucial notion in the theory of dynamical systems, since it contains information on stability and robustness properties. However, recent applications in nonautonomous bifurcation theory showed that a detailed insight into the fine structure of this spectral notion is necessary. On this basis, we explore a helpful connection between the dichotomy spectrum and operator theory. It relates the asymptotic behavior of linear nonautonomous difference equations to the point, surjectivity and Fredholm spectra of weighted shifts. This link yields several dynamically meaningful subsets of the dichotomy spectrum, which not only allows to classify and detect bifurcations, but also simplifies proofs for results on the long term behavior of difference equations with explicitly time-dependent right-hand side.

Exponential Dichotomy for Noninvertible Linear Difference Equations

“…For m < 0, suppose (1) has an exponential dichotomy on k ≤ m with projection P (k) of rank r. Then (1) has an exponential dichotomy on k ≤ 0 with projection of rank r if and only if Φ(0, m) is one to one on N P (m); (1) has an exponential dichotomy on k ≤ 0 with projection P (k) for k ≤ m if and only if Φ(0, m) is one to one on N P (m) and N (Φ(0, m)) ⊂ RP (m). Note that the first statement here was proved already by Pötzsche [14] (inspired by Lin [11]) but the second statement appears to be new.…”

“…In Section 4 we study the possibility of extending the dichotomy to a larger interval. First we show if an equation has a dichotomy on [m, ∞) with projection P (k) of rank r, where m > 0, then it can be extended to Z + if and only if dim (Φ(m, 0) −1 (RP (m)))) = r. This was proved already by Pötzsche [14] (inspired by Lin [11]) but here we show that in addition the dichotomy can be extended without changing the projection when k ≥ m. The results for Z − are rather different. For m < 0, suppose (1) has an exponential dichotomy on k ≤ m with projection P (k) of rank r. Then (1) has an exponential dichotomy on k ≤ 0 with projection of rank r if and only if Φ(0, m) is one to one on N P (m); (1) has an exponential dichotomy on k ≤ 0 with projection P (k) for k ≤ m if and only if Φ(0, m) is one to one on N P (m) and N (Φ(0, m)) ⊂ RP (m).…”

Nonuniform Dichotomy Spectrum Intervals: Theorem and Computation

“…Reducibility and normal forms for nonautonomous differential equations by using dichotomy spectrum has been given in [37,38]. For more results about dichotomy spectrum, see [24,25,32,33,36] and the references therein.…”

Nonuniform dichotomy spectrum and reducibility for nonautonomous equations

“…Among the different topics on classical exponential dichotomies, the dichotomy spectrum is very important and many results have been obtained. We refer the reader to [2,3,18,26,29,30,32,34,35] and the references therein. The definition and investigation for finite-time hyperbolicity has also been studied in [16,22,23].…”