 
Summary: Transitivity and topological mixing for C1
diffeomorphisms
Flavio Abdenur Sylvain Crovisier
July 29, 2011
To Steve Smale, for his influence on differentiable dynamics.
Abstract
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,
closing lemma.
MSC 2000: 37B20, 37C05, 37C20, 37C29, 37C50, 37D05.
1 Introduction
In his seminal dissertation about differentiable dynamical systems [22], Smale described the
recurrence of hyperbolic diffeomorphisms:
Smale's spectral decomposition theorem. Consider a diffeomorphism f of a compact man
ifold. If the nonwandering set (f) is hyperbolic and contains a dense set of periodic points
