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Periodic orbits and chain-transitive sets of C1-diffeomorphisms

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Abstract

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).

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  1. CNRS – Laboratoire Analyse, Géométrie et Applications, UMR 7539, Institut Galilée, Université Paris 13, Avenue J.-B. Clément, 93430, Villetaneuse, France

    Sylvain Crovisier

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Correspondence to Sylvain Crovisier.

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Crovisier, S. Periodic orbits and chain-transitive sets of C1-diffeomorphisms. Publ.math.IHES 104, 87–141 (2006). https://doi.org/10.1007/s10240-006-0002-4

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