Structural stability of diffeomorphisms on two-manifolds

Abstract

In [8] Robbin proved that the C2-diffeomorphisms on a compact manifold which satisfy Axiom A and the Transversality Condition are structurally stable in the C ~ sense. However, some basic results in the theory of Dynamical Systems, such as the Closing-lemma [7], are only known in the Cl-topology. Therefore, as Robbin observed, it would be of interest to extend his result for C ~ diffeomorphisms. We construct in this paper the analogue of Palis-Smale tubular families [6], for such diffeomorphisms on two dimensional manifolds. These families provide a rather fine geometric structure of the diffeomorphism near the basic sets and the relation between them. From this fact it follows the structural stability of the diffeomorphisms. We will also prove, the stability of diffeomorphisms satisfying Axiom A in a neighborhood of a basic set for two dimensional manifolds and some other cases. We will now set some notation, basic definitions, and the statements of our results. Let Diff ~(M) be the set of C~-diffeomorphisms of M with the C ttopology, where M is a compact n-dimensional manifold without boundary. If feDiffl(M), f2(f) will denote the non-wandering set of f and Per(f) the set of periodic points off. A diffeomorphism f is said to satisfy Axiom A if (i) f2(f) is hyperbolic, namely, the tangent bundle of M over t2 splits in a Df-invariant Whitney sum, TaM=ES~ E", and there exists a riemannian metric ][. II on M such that