Persistent bundles over a two dimensional compact set Pierre

The C-structurally stable diffeomorphims of a compact manifold are those that satisfy Axiom A and the strong transversality condition (AS). We generalize the concept of AS from diffeomorphisms to invariant compact subsets. Among other properties, we show the structural stability of the AS invariant compact sets K of surface diffeomorphisms f . Moreover if f̂ is the dynamics of a compact manifold, which fibers over f and such that the bundle is normally hyperbolic over the non-wandering set of f|K , then the bundle overK is persistent. This provides non trivial examples of persistent laminations that are not normally hyperbolic. A classical result states that hyperbolic compact sets are C1-structurally stable. A compact subset K of a manifold M is hyperbolic for a diffeomorphism f of M if it is invariant (f(K) = K) and the tangent bundle of M restricted to K splits into two Tf -invariant subbundles contracted and expanded respectively. An invariant subset K of a diffeomorphism f is C1-structurally stable if every C1-perturbation f ′ of f lets invariant a compact set K ′ homeomorphic to K by an embedding C0-close to the inclusion K →֒ M which conjugates the dynamics f|K and f ′ |K ′. Such a result was generalized toward two directions that we would love to unify. The first was to describe the C1-structurally stable diffeomorphisms of compact manifolds (K is then the whole manifold). This description used the so-called concept of Axiom A diffeomorphisms: the diffeomorphisms for which the non-wandering set is hyperbolic and equal to the closure of the set of periodic points. A diffeomorphism satisfies Axiom A and the strong transversality condition (AS) if moreover the stable and unstable manifolds of two non-wandering points intersect each other transversally. The works of Smale [Sma67], Palis [PS70], de Melo [dM73], Mañe [Mañ88], Robbin [Rob71] and Robinson [Rob76] have achieved a satisfactory description of the C1-structurally stable diffeomorphisms stated in the following theorem.