On the stability of weakly hyperbolic invariant sets

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 KY, which is homeomorphic to the weakly hyperbolic set K of the initial system, close to K, and the dynamics on KY 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 [16-20]. 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)=KY.