Periodic orbits and chain-transitive sets of C1-diffeomorphisms

Sylvain Crovisier (1)

1 LAGA - Laboratoire Analyse, Géométrie et Applications

We prove that the chain-transitive sets of C1-generic diffeomorphisms 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 C1-perturbation an orbit connecting several prescribed points. One deduces a weak shadowing property satisfied by C1-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).

Domaines: Systèmes dynamiques [math.DS]

Sylvain Crovisier. Periodic orbits and chain-transitive sets of C1-diffeomorphisms. Publications mathematiques de l' IHES, 2006, 104 (1), pp.87-141. ⟨10.1007/s10240-006-0002-4⟩. ⟨hal-00538119⟩