在$\RR^{p+2}$上扩展$T^p$自同构及实现DE吸引子
Extending $T^p$ automorphisms over $\RR^{p+2}$ and realizing DE attractors
摘要：
本文考虑DE吸引子通过流形的自同构的实现。对于连通的闭$p$维流形$M$的任何扩张自映射$ϕ:M\to M$，只要$q\geq p+1$是紧支集，就总能实现一个从$ϕ$导出的$(p,q)$型吸引子。因此，较低的余维实现更有趣，与稳定范围以下的打结问题有关。我们证明了对于标准光滑$p$维环面$T^p$的任何扩展自映射$ϕ$，存在$\RR^{p+2}$的紧支撑自同构，实现了从$ϕ$导出的吸引子.构造的一个关键成分是理解$T^p$的自同构，其通过标准的无结嵌入$\imath_p:T^p\hookrightarrow\RR^{p+2}$在$\RR^{p+2}$上扩展为自同构。我们证明了这些自同构构成了$\Aut(T^p)$的一个子群$E_{\imath_p}$，其指数至多为$2^p-1$ .
英文摘要：
In this paper we consider the realization of DE attractors by self-diffeomorphisms of manifolds. For any expanding self-map $ϕ:M\to M$ of a connected, closed $p$-dimensional manifold $M$, one can always realize a $(p,q)$-type attractor derived from $ϕ$ 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 $ϕ$ of a standard smooth $p$-dimensional torus $T^p$, there is compactly-supported self-diffeomorphism of $\RR^{p+2}$ realizing an attractor derived from $ϕ$. 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 $2^p-1$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
