On problems of the theory of stability of weakly hyperbolic invariant sets

Abstract

This paper represents a brief survey of the theory of stability of weakly hyperbolic invariant sets. In a series of papers published by the author together with V. A. Pliss and G. R. Sell, it was proved that a weakly hyperbolic invariant set is stable even in the absence of the Lipschitz condition. However, the question of uniqueness of leafs of a weakly hyperbolic invariant set of a perturbed system remains open. The paper shows the relationship of this problem with the so-called plaque expansivity conjecture in the theory of dynamical systems.