In the first part of this text we give a survey of the properties satisfied by the C1-generic conservative diffeomorphisms of compact surfaces. The main result that we will discuss is that a C1-generic conservative diffeomorphism of a connected compa
ct surface is transitive. It is obtain as a consequence of a connecting lemma for pseudo-orbits. In the last parts we expose some recent developments of the C1-perturbation technics and the proof of this connecting lemma. We are not aimed to deal with technicalities nor to give the finest availab
Les travaux presentes dans ce memoire portent sur la dynamique de diffeomorphismes de varietes compactes. Pour letude des proprietes generiques ou pour la construction dexemples, il est souvent utile de savoir perturber un syst`eme. Ceci soul`eve gen
eralement des probl`emes delicats : 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 transversalite, connexion dorbites, perturbation de la dynamique tangente, realisation dextensions... Nous en tirons diverses applications `a la description de la dynamique des diffeomorphismes C^1-generiques. <p> 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.
We prove that any diffeomorphism of a compact manifold can be C^1-approximated by a diffeomorphism which exhibits a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by a diffeomorphism which is partially hyperbolic (its
chain-recurrent set splits into partially hyperbolic pieces whose centre bundles have dimensions less or equal to two). We also study in a more systematic way the central models introduced in arXiv:math/0605387.
We study the ergodic theory of non-conservative C^1-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show that generic
(for the weak topology) ergodic measures of C^1-generic diffeomorphisms are nonuniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set L of any C^1-generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set L. In addition, confirming a claim made by R. Mane in 1982, we show that hyperbolic measures whose Oseledets splittings are dominated satisfy Pesins Stable Manifold Theorem, even if the diffeomorphism is only C^1.
Answering a question of Smale, we prove that the space of C1 diffeomorphisms of a compact manifold contains a residual subset of diffeomorphisms whose centralizers are trivial.
We prove that the spaces of C1 symplectomorphisms and of C1 volume-preserving diffeomorphisms of connected manifolds both contain residual subsets of diffeomorphisms whose centralizers are trivial. (Les diffeomorphismes conservatifs C1-generiques ont
un centralisateur trivial. Nous montrons que lespace des symplectomorphismes de classe C1 et lespace des diffeomomorphismes de classe C1 preservant une forme volume contiennent tous deux des sous-ensembles residuels de diffeomorphismes dont le centralisateur est trivial.)
Given any compact manifold M, we construct a non-empty open subset O of the space of C^1-diffeomorphisms of M and a dense subset D of O such that the centralizer of every diffeomorphism in D is uncountable, hence non-trivial.
We construct different types of quasiperiodically forced circle homeomorphisms with transitive but non-minimal dynamics. Concerning the recent Poincare-like classification for this class of maps of Jaeger-Stark, we demonstrate that transitive but non
-minimal behaviour can occur in each of the different cases. This closes one of the last gaps in the topological classification. Actually, we are able to get some transitive quasiperiodically forced circle homeomorphisms with rather complicated minimal sets. For example, we show that, in some of the examples we construct, the unique minimal set is a Cantor set and its intersection with each vertical fibre is uncountable and nowhere dense (but may contain isolated points). We also prove that minimal sets of the later kind cannot occur when the dynamics are given by the projective action of a quasiperiodic SL(2,R)-cocycle. More precisely, we show that, for a quasiperiodic SL(2,R)-cocycle, any minimal strict subset of the torus either is a union of finitely many continuous curves, or contains at most two points on generic fibres.
