A finite-time condition for exponential dichotomy

Abstract

We prove that if a difference equation on Z has an exponential dichotomy on sufficiently many intervals of sufficiently large fixed length and with uniform constants and exponents, then it has an exponential dichotomy on Z. We give applications of the result to slowly varying systems, almost periodic systems, a perturbation theorem, the variational equation along pseudo orbits lying in a hyperbolic set of a diffeomorphism, transversality of homoclinic orbits of a diffeomorphism and hyperbolicity of a compact invariant set of a diffeomorphism.

Keywords:

• linear difference equation
• exponential dichotomy
• diffeomorphism
• hyperbolic

AMS Subject Classification::

• 34D09
• 39A10

Acknowledgement

K.J.P. was partially supported by NSC (Taiwan) 97-2115-M-002-011-MY2.