# American Institute of Mathematical Sciences

April  2013, 33(4): 1297-1311. doi: 10.3934/dcds.2013.33.1297

## Admissibility versus nonuniform exponential behavior for noninvertible cocycles

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

Received  October 2011 Revised  January 2012 Published  October 2012

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.
Citation: Luis Barreira, Claudia Valls. Admissibility versus nonuniform exponential behavior for noninvertible cocycles. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1297-1311. doi: 10.3934/dcds.2013.33.1297
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##### References:
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