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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

Fan Ding 1, , Yi Liu 2, , Shicheng Wang 1, and  Jiangang Yao 2,

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 $2^p-1$.
Keywords: DE attractors, extendable subgroup..
Mathematics Subject Classification: Primary: 37C70; Secondary: 57R4.
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

[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

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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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