  On the one hand, we prove that the spaces of C^1 symplectomorphisms and of
C^1 volume-preserving diffeomorphisms both contain residual subsets of
diffeomorphisms whose centralizers are trivial. On the other hand, we show that
the space of C^1 diffeomorphisms of the circle and a non-empty open set of C^1
diffeomorphisms of the two-sphere contain dense subsets of diffeomorphisms
whose centralizer has a sub-group isomorphic to R.
