Admissibility versus nonuniform exponential behavior for noninvertible cocycles

Luis Barreira; Claudia Valls

doi:10.3934/dcds.2013.33.1297

Article Contents

2013, Volume 33, Issue 4: 1297-1311. Doi: 10.3934/dcds.2013.33.1297

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Admissibility versus nonuniform exponential behavior for noninvertible cocycles

Luis Barreira^1, and
Claudia Valls^2,

1.
Departamento de Matemática, Instituto Superior Técnico, UTL, 1049-001 Lisboa
2.
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa

Received: September 30, 2011

Revised: December 31, 2011

Published: March 2013

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Abstract

We study the relation between the notions of exponential dichotomy and admissibility for a nonautonomous dynamics with discrete time. More precisely, we consider $\mathbb{Z}$-cocycles defined by a sequence of linear operators in a Banach space, and we give criteria for the existence of an exponential dichotomy in terms of the admissibility of the pairs $(\ell^p,\ell^q)$ of spaces of sequences, with $p\le q$ and $(p,q)\ne(1,\infty)$. We extend the existing results in several directions. Namely, we consider the general case of nonuniform exponential dichotomies; we consider $\mathbb{Z}$-cocycles and not only $\mathbb{N}$-cocycles; and we consider exponential dichotomies that need not be invertible in the stable direction. We also exhibit a collection of admissible pairs of spaces of sequences for any nonuniform exponential dichotomy.

Keywords:

Admissibility,
nonuniform exponential dichotomies.

Mathematics Subject Classification: Primary: 34D09, 37D25.

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