Generalizations of the notion of hyperbolicity

Abstract

We describe several generalizations of the classical notion of hyperbolicity for a sequence of linear mappings. It is shown that the following three statements are equivalent: (i) the corresponding linear non-homogeneous system has a bounded solution for any bounded nonhomogeneity, (ii) the sequence has a (C, λ)-structure, (iii) the sequence is piecewise hyperbolic with long enough intervals of hyperbolicity.

Keywords:

• Dynamical systems
• Hyperbolicity
• Transversality
• Bounded solutions

Keywords:

• 37D05

Acknowledgements

The author is deeply grateful to the referee for useful comments which helped to improve the presentation. This research was supported by the Russian Foundation for Basic Research (project 05-01-01079) and by the Program “Leading Scientific Schools” (project 2271.2003.1).