 
Summary: ON THE REGULARIZATION OF CONSERVATIVE MAPS
ARTUR AVILA
Abstract. We show that smooth maps are C1dense 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 nonexistence 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
