Nonuniform hyperbolicity for C^1-generic diffeomorphisms

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.

The C1 generic diffeomorphism has trivial centralizer

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.

C1-generic conservative diffeomorphisms have trivial centralizer

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.)

Local density of diffeomorphisms with large centralizers

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.