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Summary: ON THE REGULARIZATION OF CONSERVATIVE MAPS
ARTUR AVILA
Abstract. We show that smooth maps are C1-dense among C1 volume pre-
serving maps.
1. Introduction
Let M and N be C
manifolds1
and let Cr
(M, N) (r N{}) be the space of
Cr
maps from M to N, endowed with the Whitney topology. It is a well known fact
that C
maps are dense in Cr
(M, N). Such a result is very useful in differentiable
topology and in dynamical systems (as we will discuss in more detail). On the other
hand, in closely related contexts, it is the non-existence of a regularization theorem
that turns out to be remarkable: if homeomorphisms could always be approximated
by diffeomorphisms then the whole theory of exotic structures would not exist.
Palis and Pugh [20] seem to have been the first to ask about the corresponding
regularization results in the case of conservative and symplectic maps. Here one
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