Summary: Global dominated splittings and the C 1 Newhouse phenomenon
Flavio Abdenur, Christian Bonatti, and Sylvain Crovisier
April 15, 2004
We prove that given a compact ndimensional boundaryless manifold M , n # 2, there
exists a residual subset R of the space of C 1 di#eomorphisms Di# 1 (M) such that given
any chaintransitive set K of f # R then either K admits a dominated splitting or else
K is contained in the closure of an infinite number of periodic sinks/sources. This result
generalizes the generic dichotomy for homoclinic classes in [BDP].
It follows from the above result that given a C 1 generic di#eomorphism f then either the
set# f) may be decomposed into a finite number of pairwise disjoint compact
sets each of which admits a dominated splitting, or else f exhibits infinitely many periodic
sinks/sources (the ``C 1 Newhouse phenomenon''). This result answers a question in [BDP]
and generalizes the generic dichotomy for surface di#eomorphisms in [M].
1 Context and notation
Throughout this paper, M denotes a compact boundaryless manifold of dimension n # 2 and
Di# 1 (M) is the space of C 1 di#eomorphisms on M with the usual topology.
Given an open subset U of Di# 1 (M ), a subset R of U is residual in U if R contains the
intersection of a countable family of open and dense subsets of U ; in this case R is dense in U .