Extending $T^p$ automorphisms over ${\text{RR}}^{p+2}$ and realizing DE attractors

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