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

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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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     author = {Crovisier, Sylvain},
     title = {Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {87--141},
     publisher = {Springer},
     volume = {104},
     year = {2006},
     doi = {10.1007/s10240-006-0002-4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10240-006-0002-4/}
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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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