Periodic orbits and chain-transitive sets of C 1 -diffeomorphisms
Publications Mathématiques de l'IHÉS, Volume 104  (2006), p. 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).

@article{PMIHES_2006__104__87_0,
     author = {Crovisier, Sylvain},
     title = {Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     publisher = {Springer},
     volume = {104},
     year = {2006},
     pages = {87-141},
     doi = {10.1007/s10240-006-0002-4},
     zbl = {pre05117095},
     language = {en},
     url = {http://www.numdam.org/item/PMIHES_2006__104__87_0}
}
Crovisier, Sylvain. Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms. Publications Mathématiques de l'IHÉS, Volume 104 (2006) , pp. 87-141. doi : 10.1007/s10240-006-0002-4. http://www.numdam.org/item/PMIHES_2006__104__87_0/

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