Periodic orbits and chain-transitive sets of C1-diffeomorphisms

We prove that the chain-transitive sets of C-1-generic dilleomorphisms are approximated in the Hausdorff topology by periodic orbits. This implies that the homoclinic classes are dense among the chain-recurrence classes. This result is a consequence of a global connecting lemma, which allows to build by a C-1-perturbation an orbit connecting several prescribed points. One deduces a weak shadowing property satisfied by C-1-generic diffeomorphisms: any pseudo-orbit is approximated in the Hausdorff topology by a finite segment of a genuine orbit. As a consequence, we obtain a criterion for proving the tolerance stability conjecture in Diff(1) (M).