  In this paper we consider the realization of DE attractors by
self-diffeomorphisms of manifolds. For any expanding self-map $\phi:M\to M$ of
a connected, closed $p$-dimensional manifold $M$, one can always realize a
$(p,q)$-type attractor derived from $\phi$ by a compactly-supported
self-diffeomorphsm of $\RR^{p+q}$, as long as $q\geq p+1$. Thus lower
codimensional realizations are more interesting, related to the knotting
problem below the stable range. We show that for any expanding self-map $\phi$
of a standard smooth $p$-dimensional torus $T^p$, there is compactly-supported
self-diffeomorphism of $\RR^{p+2}$ realizing an attractor derived from $\phi$.
A key ingredient of the construction is to understand automorphisms of $T^p$
which extend over $\RR^{p+2}$ as a self-diffeomorphism via the standard
unknotted embedding $\imath_p:T^p\hookrightarrow\RR^{p+2}$. We show that these
automorphisms form a subgroup $E_{\imath_p}$ of $\Aut(T^p)$ of index at most
$2^p-1$.
