Periodic orbits and chain-transitive sets of C 1 -diffeomorphisms
Publications Mathématiques de l'IHÉS, Tome 104 (2006), pp. 87-141

We prove that the chain-transitive sets of C 1 -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 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).

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     title = {Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms},
     journal = {Publications Math\'ematiques de l'IH\'ES},
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Crovisier, Sylvain. Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms. Publications Mathématiques de l'IHÉS, Tome 104 (2006), pp. 87-141. doi: 10.1007/s10240-006-0002-4

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