We prove that the chain-transitive sets of -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 -perturbation an orbit connecting several prescribed points. One deduces a weak shadowing property satisfied by -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 .
@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}, pages = {87--141}, publisher = {Springer}, volume = {104}, year = {2006}, doi = {10.1007/s10240-006-0002-4}, language = {en}, url = {http://archive.numdam.org/articles/10.1007/s10240-006-0002-4/} }
TY - JOUR AU - Crovisier, Sylvain TI - Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms JO - Publications Mathématiques de l'IHÉS PY - 2006 SP - 87 EP - 141 VL - 104 PB - Springer UR - http://archive.numdam.org/articles/10.1007/s10240-006-0002-4/ DO - 10.1007/s10240-006-0002-4 LA - en ID - PMIHES_2006__104__87_0 ER -
%0 Journal Article %A Crovisier, Sylvain %T Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms %J Publications Mathématiques de l'IHÉS %D 2006 %P 87-141 %V 104 %I Springer %U http://archive.numdam.org/articles/10.1007/s10240-006-0002-4/ %R 10.1007/s10240-006-0002-4 %G en %F 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://archive.numdam.org/articles/10.1007/s10240-006-0002-4/
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