May  2012, 32(5): 1639-1655. doi: 10.3934/dcds.2012.32.1639

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

1. 

School of Mathematical Sciences, Peking University, Beijing, 100871, China, China

2. 

Department of Mathematics, 970 Evans Hall, University of California, Berkeley, CA 94720-3840, United States, United States

Received  December 2010 Revised  September 2011 Published  January 2012

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 $2^p-1$.
Citation: Fan Ding, Yi Liu, Shicheng Wang, Jiangang Yao. Extending $T^p$ automorphisms over $\mathbb{R}^{p+2}$ and realizing DE attractors. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1639-1655. doi: 10.3934/dcds.2012.32.1639
References:
[1]

Hans G. Bothe, The ambient structure of expanding attractors. II. Solenoids in 3-manifolds,, Math. Nachr., 112 (1983), 69.  doi: 10.1002/mana.19831120105.  Google Scholar

[2]

Fan Ding, Yi Liu, Shicheng Wang and Jiangang Yao, Spin structure and codimension-two homeomorphism extension,, preprint, ().   Google Scholar

[3]

Fan Ding, Jianzhong Pan, Shicheng Wang and Jiangang Yao, Only rational homology spheres admit $\Omega(f)$ to be union of DE attractors,, Ergodic Theory Dynam. Systems, 30 (2010), 1399.  doi: 10.1017/S014338570900073X.  Google Scholar

[4]

Karel Dekimpe, Michał Sadowski and Andrzej Szczepański, Spin structures on flat manifolds,, Monatsh. Math., 148 (2006), 283.   Google Scholar

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David Epstein and Michael Shub, Expanding endomorphisms of flat manifolds,, Topol., 7 (1968), 139.  doi: 10.1016/0040-9383(68)90022-0.  Google Scholar

[6]

Mikhael Gromov, Groups of polynomial growth and expanding maps,, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53.   Google Scholar

[7]

André Haefliger, Plongements différentiables dans le domaine stable, (French),, Comment. Math. Helv., 37 (): 155.   Google Scholar

[8]

Morris W. Hirsch, "Differential Topology,", Graduate Texts in Mathematics, (1976).   Google Scholar

[9]

Boju Jiang, Yi Ni and Shicheng Wang, 3-manifolds that admit knotted solenoids as attractors,, Trans. Amer. Math. Soc., 356 (2004), 4371.  doi: 10.1090/S0002-9947-04-03503-2.  Google Scholar

[10]

Boju Jiang, Shicheng Wang and Hao Zheng, No embeddings of solenoids into surfaces,, Proc. Amer. Math. Soc., 136 (2008), 3697.  doi: 10.1090/S0002-9939-08-09340-4.  Google Scholar

[11]

John W. Milnor and James D. Stasheff, "Characteristic Classes,", Annals of Mathematics Studies, (1974).   Google Scholar

[12]

Morris Newman, "Integral Matrices,", Pure and Applied Mathematics, (1972).   Google Scholar

[13]

Stephen Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

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Thomas E. Stewart, On groups of diffeomorphisms,, Proc. Amer. Math. Soc., 11 (1960), 559.  doi: 10.1090/S0002-9939-1960-0120651-6.  Google Scholar

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Wen-tsün Wu, On the isotopy of $C^r$-manifolds of dimension $n$ in euclidean $(2n+1)$-space,, Sci. Record (N.S.), 2 (1958), 271.   Google Scholar

show all references

References:
[1]

Hans G. Bothe, The ambient structure of expanding attractors. II. Solenoids in 3-manifolds,, Math. Nachr., 112 (1983), 69.  doi: 10.1002/mana.19831120105.  Google Scholar

[2]

Fan Ding, Yi Liu, Shicheng Wang and Jiangang Yao, Spin structure and codimension-two homeomorphism extension,, preprint, ().   Google Scholar

[3]

Fan Ding, Jianzhong Pan, Shicheng Wang and Jiangang Yao, Only rational homology spheres admit $\Omega(f)$ to be union of DE attractors,, Ergodic Theory Dynam. Systems, 30 (2010), 1399.  doi: 10.1017/S014338570900073X.  Google Scholar

[4]

Karel Dekimpe, Michał Sadowski and Andrzej Szczepański, Spin structures on flat manifolds,, Monatsh. Math., 148 (2006), 283.   Google Scholar

[5]

David Epstein and Michael Shub, Expanding endomorphisms of flat manifolds,, Topol., 7 (1968), 139.  doi: 10.1016/0040-9383(68)90022-0.  Google Scholar

[6]

Mikhael Gromov, Groups of polynomial growth and expanding maps,, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53.   Google Scholar

[7]

André Haefliger, Plongements différentiables dans le domaine stable, (French),, Comment. Math. Helv., 37 (): 155.   Google Scholar

[8]

Morris W. Hirsch, "Differential Topology,", Graduate Texts in Mathematics, (1976).   Google Scholar

[9]

Boju Jiang, Yi Ni and Shicheng Wang, 3-manifolds that admit knotted solenoids as attractors,, Trans. Amer. Math. Soc., 356 (2004), 4371.  doi: 10.1090/S0002-9947-04-03503-2.  Google Scholar

[10]

Boju Jiang, Shicheng Wang and Hao Zheng, No embeddings of solenoids into surfaces,, Proc. Amer. Math. Soc., 136 (2008), 3697.  doi: 10.1090/S0002-9939-08-09340-4.  Google Scholar

[11]

John W. Milnor and James D. Stasheff, "Characteristic Classes,", Annals of Mathematics Studies, (1974).   Google Scholar

[12]

Morris Newman, "Integral Matrices,", Pure and Applied Mathematics, (1972).   Google Scholar

[13]

Stephen Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[14]

Thomas E. Stewart, On groups of diffeomorphisms,, Proc. Amer. Math. Soc., 11 (1960), 559.  doi: 10.1090/S0002-9939-1960-0120651-6.  Google Scholar

[15]

Wen-tsün Wu, On the isotopy of $C^r$-manifolds of dimension $n$ in euclidean $(2n+1)$-space,, Sci. Record (N.S.), 2 (1958), 271.   Google Scholar

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