
Mathematics 2008
Extending $T^p$ automorphisms over $\RR^{p+2}$ and realizing DE attractorsAbstract: In this paper we consider the realization of DE attractors by selfdiffeomorphisms of manifolds. For any expanding selfmap $\phi:M\to M$ of a connected, closed $p$dimensional manifold $M$, one can always realize a $(p,q)$type attractor derived from $\phi$ by a compactlysupported selfdiffeomorphsm 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 selfmap $\phi$ of a standard smooth $p$dimensional torus $T^p$, there is compactlysupported selfdiffeomorphism 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 selfdiffeomorphism 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 $2^p1$.
