Periodic orbits and chain-transitive sets of C1-diffeomorphisms



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 Diff1(M).


Periodic Orbit Periodic Point Hausdorff Distance Hausdorff Topology Forward Orbit 


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Copyright information

© IHES and Springer-Verlag 2006

Authors and Affiliations

  1. 1.CNRS – Laboratoire Analyse, Géométrie et Applications, UMR 7539, Institut GaliléeUniversité Paris 13VilletaneuseFrance

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