Perturbation de la dynamique de difféomorphismes en topologie C^1 / Perturbation of the dynamics of diffeomorphisms in the C^1-topology

Details Perturbation de la dynamique de difféomorphismes en topologie C^1 / Perturbation of the dynamics of diffeomorphisms in the C^1-topology Perturbation de la dynamique de difféomorphismes en topologie C^1 / Perturbation of the dynamics of diffeomorphisms in the C^1-topology

2009-12-15

arxiv.org/abs/0912.2896v1

Mathematics

Dynamical Systems

Scientific paper

Les travaux pr\'esent\'es dans ce m\'emoire portent sur la dynamique de diff\'eomorphismes de vari\'et\'es compactes. Pour l'\'etude des propri\'et\'es g\'en\'eriques ou pour la construction d'exemples, il est souvent utile de savoir perturber un syst\`eme. Ceci soul\`eve g\'en\'eralement des probl\`emes d\'elicats : une modification locale de la dynamique peut engendrer un changement brutal du comportement des orbites. En topologie C^1, nous proposons diverses techniques permettant de perturber tout en contr\^olant la dynamique : mise en transversalit\'e, connexion d'orbites, perturbation de la dynamique tangente, r\'ealisation d'extensions... Nous en tirons diverses applications \`a la description de la dynamique des diff\'eomorphismes C^1-g\'en\'eriques.

This memoir deals with the dynamics of diffeomorphisms of compact manifolds. For the study of generic properties or for the construction of examples, it is often useful to be able to perturb a system. This generally leads to delicate problems: a local modification of the dynamic may cause a radical change in the behavior of the orbits. For the C^1 topology, we propose various techniques which allow to perturb while controlling the dynamic: setting in transversal position, connection of orbits, perturbation of the tangent dynamics,... We derive various applications to the description of the C^1-generic diffeomorphisms.

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