Persistent bundles over a two dimensional compact set

The $C^1$-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 $\widehat{f}$ 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 $f_{\mathrm{\mid }K}$, then the bundle over $K$ is persistent. This provides non trivial examples of persistent laminations that are not normally hyperbolic.