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Transitivity and topological mixing for C1 diffeomorphisms

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,
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 non-wandering set (f) is hyperbolic and contains a dense set of periodic points


Source: Abdenur, Flavio - Departamento de Matemática, Pontifícia Universidade Católica do Rio Grande do Sul


Collections: Mathematics