On two notions of exponential dichotomy

摘要

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.