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Summary: Non-wandering sets with non-empty interior
F. Abdenur, C. Bonatti and L.J. Daz
March 19, 2003
Abstract
We study di eomorphisms of a closed connected manifold whose non-wandering set has
non-empty interior and conjecture that C 1 -generic di eomorphisms whose non-wandering
set has non-empty interior are transitive. We rst prove this conjecture in three cases:
hyperbolic, partially hyperbolic with two hyperbolic bundles, and tame di eomorphisms (in
the rst case, the conjecture is a folklore result and in the second one it follows adapting the
proof in [B]).
We study this conjecture without global assumptions and prove that, generically, a ho-
moclinic class with non-empty interior is either the whole manifold or else accumulated by
in nitely many di erent homoclinic classes. Finally, we prove that, generically, homoclinic
classes and non-wandering sets with non-empty interior are weakly hyperbolic (existence of
a dominated or a volume hyperbolic splitting).
1 Introduction
When the dynamics of a system (here a di eomorphism f) is complicated, one tries to study
the global dynamics of the system by looking at those regions of the ambient space which con-
centrate the recurrences and the intricate parts of the dynamics. The non-wandering set is one
of the most common sets used by dynamicists for localizing the complexity (other possibilities
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