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 $\hat f$ 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 $f_{|K}$, then the bundle over $K$ is persistent. This provides non trivial examples of persistent laminations that are not normally hyperbolic.Comment: 32 p. The proof is much more (10 p.) detailed than previousl

Generic family with robustly infinitely many sinks

We show, for every $r>d\ge 0$ or $r=d\ge 2$, the existence of a Baire generic set of $C^d$-families of $C^r$-maps $(f_a)_{a\in (-1,1)^k}$ of a manifold $M$ of dimension $\ge 2$, so that for every $a$ small the map $f_a$ has infinitely many sinks. When the dimension of the manifold is greater than $3$, the generic set is formed by families of diffeomorphisms. This result is a counter-example to a conjecture of Pugh and Shub.Comment: to appear in Inventiones mathematica

Lectures on Structural Stability in Dynamics

These lectures present results and problems on the characterization of structurally stable dynamics. We will shed light those which do not seem to depend on the regularity class (holomorphic or differentiable). Furthermore, we will present some links between the problems of structural stability in dynamical systems and in singularity theory.Comment: 30pages and 9 figure