On two notions of exponential dichotomy

ABSTRACT

In this paper, we propose a weaker form of hyperbolicity in which the conditional stability along the stable and unstable directions is allowed to be non-uniform depending on each particular vector, instead of on each invariant space. We also give explicit examples of the notion. Moreover, we show that the size of the set of vectors for which the spoiling of the uniform exponential behaviour has a prescribed bound, measured in terms of the Lebesgue measure on the unit sphere, may grow to infinity with any given speed.

Keywords:

• Exponential dichotomies
• sequences of matrices
• stability

2010 Mathematics Subject Classification.:

• Primary: 37D99

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Partially supported by FCT/Portugal [project number UID/MAT/04459/2013].