Nicolas Gourmelon

Local Cr Dynamics

f is a diffeomorphism of a compact manifold, 0< r <infty and R(p) is the chain-recurrent class of a saddle point p.

We call "local" a chain-recurrent set included in the union of a finite number of hyperbolic sets and compact parts of their respective stable and unstable laminations. Such a set may be conveniently represented by a graph. Until now, all known generic obstructions to dominated splittings are local, that is, are visible on local sets.

We propose classifying tools for the dynamics that may appear by Cr-perturbations close to local sets. Among other consequences, we get sharp dichotomies between local volume hyperbolicity on R(p), and local Cr Newhouse phenomenon, and between local stable/unstable dominated splitting and homoclinic tangencies.

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