Tempered exponential dichotomies: admissibility and stability under perturbations. Issue 4 (1st October 2016)

• Record Type: Journal Article
• Authors: Barreira, Luis; Dragičević, Davor; Valls, Claudia
• Is Part Of: Dynamical systems. Volume 31:Issue 4(2016)
• Page Start: 525
• Page End: 545
• Publication Date: 2016-10-01
• Subjects: Tempered exponential dichotomies -- robustness -- conjugacies; Primary: 37D99; Differentiable dynamical systems -- Periodicals; 515.35205
• DOI: 10.1080/14689367.2016.1159663
• Languages: English
• ISSNs: 1468-9367

Abstract

ABSTRACT: We give a characterization of the notion of a tempered exponential dichotomy on a Banach space in terms of an admissibility property. We note that for a linear cocycle over a measure-preserving transformation satisfying a certain integrability assumption, it follows from the multiplicative ergodic theorem that the dynamics admits a tempered exponential dichotomy if and only if all Lyapunov exponents are non-zero almost everywhere. As a consequence of our approach, we give a new proof of the robustness property of the notion of a tempered exponential dichotomy under sufficiently small linear perturbations and we establish a version of the Grobman–Hartman theorem yielding the existence of topological conjugacies between a linear dynamics with a tempered exponential dichotomy and any sufficiently small nonlinear perturbation. In addition, we show that the conjugacy maps vary continuously with the perturbation.