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

TL;DR

This paper demonstrates how to realize certain dynamical attractors derived from expanding maps on tori as self-diffeomorphisms of Euclidean spaces, focusing on low codimension cases related to knotting problems.

Contribution

It extends the understanding of automorphisms of tori that can be realized over Euclidean spaces, especially in the context of attracting sets and their embeddings.

Findings

Realization of DE attractors by self-diffeomorphisms in Euclidean spaces.

Identification of a subgroup of automorphisms extendable over 12.

Lower codimension realizations linked to knotting problems.

Abstract

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 $2^p-1$.