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

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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

Abstract
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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 and Continuous Dynamical Systems, 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), 69-102. doi: 10.1002/mana.19831120105.
[2]	Fan Ding, Yi Liu, Shicheng Wang and Jiangang Yao, Spin structure and codimension-two homeomorphism extension,, preprint, ().
[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), 1399-1417. doi: 10.1017/S014338570900073X.
[4]	Karel Dekimpe, Michał Sadowski and Andrzej Szczepański, Spin structures on flat manifolds, Monatsh. Math., 148 (2006), 283-296.
[5]	David Epstein and Michael Shub, Expanding endomorphisms of flat manifolds, Topol., 7 (1968), 139-141. doi: 10.1016/0040-9383(68)90022-0.
[6]	Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53-73.
[7]	André Haefliger, Plongements différentiables dans le domaine stable, (French),, Comment. Math. Helv., 37 (): 155.
[8]	Morris W. Hirsch, "Differential Topology," Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976.
[9]	Boju Jiang, Yi Ni and Shicheng Wang, 3-manifolds that admit knotted solenoids as attractors, Trans. Amer. Math. Soc., 356 (2004), 4371-4382. doi: 10.1090/S0002-9947-04-03503-2.
[10]	Boju Jiang, Shicheng Wang and Hao Zheng, No embeddings of solenoids into surfaces, Proc. Amer. Math. Soc., 136 (2008), 3697-3700. doi: 10.1090/S0002-9939-08-09340-4.
[11]	John W. Milnor and James D. Stasheff, "Characteristic Classes," Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1974.
[12]	Morris Newman, "Integral Matrices," Pure and Applied Mathematics, Vol. 45, Academic Press, New York-London, 1972.
[13]	Stephen Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1.
[14]	Thomas E. Stewart, On groups of diffeomorphisms, Proc. Amer. Math. Soc., 11 (1960), 559-563. doi: 10.1090/S0002-9939-1960-0120651-6.
[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), 271-275.

show all references

References:

[1]	Hans G. Bothe, The ambient structure of expanding attractors. II. Solenoids in 3-manifolds, Math. Nachr., 112 (1983), 69-102. doi: 10.1002/mana.19831120105.
[2]	Fan Ding, Yi Liu, Shicheng Wang and Jiangang Yao, Spin structure and codimension-two homeomorphism extension,, preprint, ().
[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), 1399-1417. doi: 10.1017/S014338570900073X.
[4]	Karel Dekimpe, Michał Sadowski and Andrzej Szczepański, Spin structures on flat manifolds, Monatsh. Math., 148 (2006), 283-296.
[5]	David Epstein and Michael Shub, Expanding endomorphisms of flat manifolds, Topol., 7 (1968), 139-141. doi: 10.1016/0040-9383(68)90022-0.
[6]	Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53-73.
[7]	André Haefliger, Plongements différentiables dans le domaine stable, (French),, Comment. Math. Helv., 37 (): 155.
[8]	Morris W. Hirsch, "Differential Topology," Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976.
[9]	Boju Jiang, Yi Ni and Shicheng Wang, 3-manifolds that admit knotted solenoids as attractors, Trans. Amer. Math. Soc., 356 (2004), 4371-4382. doi: 10.1090/S0002-9947-04-03503-2.
[10]	Boju Jiang, Shicheng Wang and Hao Zheng, No embeddings of solenoids into surfaces, Proc. Amer. Math. Soc., 136 (2008), 3697-3700. doi: 10.1090/S0002-9939-08-09340-4.
[11]	John W. Milnor and James D. Stasheff, "Characteristic Classes," Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1974.
[12]	Morris Newman, "Integral Matrices," Pure and Applied Mathematics, Vol. 45, Academic Press, New York-London, 1972.
[13]	Stephen Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1.
[14]	Thomas E. Stewart, On groups of diffeomorphisms, Proc. Amer. Math. Soc., 11 (1960), 559-563. doi: 10.1090/S0002-9939-1960-0120651-6.
[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), 271-275.

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