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

Fan Ding; Yi Liu; Shicheng Wang; Jiangang Yao

doi:10.3934/dcds.2012.32.1639

Article Contents

2012, Volume 32, Issue 5: 1639-1655. Doi: 10.3934/dcds.2012.32.1639

This issue Previous Article The center--focus problem and small amplitude limit cycles in rigid systems Next Article Minimal skew products with hypertransitive or mixing properties

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

1.
School of Mathematical Sciences, Peking University, Beijing, 100871
2.
Department of Mathematics, 970 Evans Hall, University of California, Berkeley, CA 94720-3840

Received: November 30, 2010

Revised: August 31, 2011

Published: April 2012

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Abstract

We show that for every expanding self-map of a connected, closed $p$-dimensional manifold $M$, and for every codimension $q\geq p+1$, there exists a corresponding $(p,q)$-type DE attractor realized by a compactly-supported self-diffeomorphsm of $\mathbb{R}^{p+q}$. Moreover, when $M$ is the standard smooth $p$-dimensional torus $T^p$, the codimension $q$ can be taken as two. As a key ingredient of the construction, for the standard unknotted embedding $\imath_p:T^p\hookrightarrow\mathbb{R}^{p+2}$, we show the automorphisms that diffeomorphically extend over $\mathbb{R}^{p+2}$ form a subgroup of $Aut(T^p)$ of index at most $2^p-1$.

Keywords:

DE attractors,
extendable subgroup.

Mathematics Subject Classification: Primary: 37C70; Secondary: 57R40.

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