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

Abstract

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 $\RR^{p+q}$, 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 $\RR^{p+2}$ realizing an attractor derived from $\varphi$. A key ingredient of the construction is to understand automorphisms of $T^p$ which extend over $\RR^{p+2}$ 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 $E_{{ı}_p}$ of $\Aut(T^p)$ of index at most $2^p-1$.

19 pages, 4 figures

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• On diffeomorphisms over surfaces trivially embedded in the 4-sphere
• No embeddings of solenoids into surfaces
• Spin structures and codimension-two homeomorphism extensions

Cited by in corpus (1)

• Spin structures and codimension-two homeomorphism extensions