On the stability of weakly hyperbolic invariant sets

Abstract The dynamical object which we study is a compact invariant set with a suitable hyperbolic structure. Stability of weakly hyperbolic sets was studied by V. A. Pliss and G. R. Sell (see [1] , [2] ). They assumed that the neutral, unstable and stable linear spaces of the corresponding linearized systems satisfy Lipschitz condition. They showed that if a perturbation is small, then the perturbed system has a weakly hyperbolic set K Y , which is homeomorphic to the weakly hyperbolic set K of the initial system, close to K , and the dynamics on K Y is close to the dynamics on K . At the same time, it is known that the Lipschitz property is too strong in the sense that the set of systems without this property is generic. Hence, there was a need to introduce new methods of studying stability of weakly hyperbolic sets without Lipschitz condition. These new methods appeared in [16] , [17] , [18] , [19] , [20] . They were based on the local coordinates introduced in [18] and the continuous on the whole weakly hyperbolic set coordinates introduced in [19] . In this paper we will show that even without Lipschitz condition there exists a continuous mapping h such that h ( K ) = K Y .