Summary: Transitivity and topological mixing for C1
Flavio Abdenur Sylvain Crovisier
July 29, 2011
To Steve Smale, for his influence on differentiable dynamics.
We prove that, on connected compact manifolds, both C1
-generic conservative diffeomor-
phisms and C1
-generic transitive diffeomorphisms are topologically mixing. This is obtained
through a description of the periods of a homoclinic class and by a control of the period of
the periodic points given by the closing lemma.
Key words: Homoclinic class, hyperbolic diffeomorphism, transitivity, topological mixing,
MSC 2000: 37B20, 37C05, 37C20, 37C29, 37C50, 37D05.
In his seminal dissertation about differentiable dynamical systems , Smale described the
recurrence of hyperbolic diffeomorphisms:
Smale's spectral decomposition theorem. Consider a diffeomorphism f of a compact man-
ifold. If the non-wandering set (f) is hyperbolic and contains a dense set of periodic points