2010, 4(3): 517-548. doi: 10.3934/jmd.2010.4.517

Nonexpanding attractors: Conjugacy to algebraic models and classification in 3-manifolds

1. 

Department of Mathematics, Tufts University, Medford, MA 02155, United States

Received  January 2010 Revised  July 2010 Published  October 2010

We prove a result motivated by Williams's classification of expanding attractors and the Franks--Newhouse Theorem on codimension-$1$ Anosov diffeomorphisms: If $\Lambda$ is a topologically mixing hyperbolic attractor such that $\dim\E^u$|$\Lambda$ = 1, then either $\Lambda$ is expanding or is homeomorphic to a compact abelian group (a toral solenoid). In the latter case, $f$|$\Lambda$ is conjugate to a group automorphism. As a corollary, we obtain a classification of all $2$-dimensional basic sets in $3$-manifolds. Furthermore, we classify all topologically mixing hyperbolic attractors in $3$-manifolds in terms of the classically studied examples, answering a question of Bonatti in [1].
Citation: Aaron W. Brown. Nonexpanding attractors: Conjugacy to algebraic models and classification in 3-manifolds. Journal of Modern Dynamics, 2010, 4 (3) : 517-548. doi: 10.3934/jmd.2010.4.517
References:
[1]

C. Bonatti, Problem in dynamical systems,, , (1999).

[2]

H. G. Bothe, Expanding attractors with stable foliations of class $C^0$,, in, 1514 (1992), 36.

[3]

B. Brenken, The local product structure of expansive automorphisms of solenoids and their associated $C^$*-algebras,, Canad. J. Math., 48 (1996), 692.

[4]

A. Brown, Constraints on dynamics preserving certain hyperbolic sets,, Ergodic Theory Dynam. Systems, ().

[5]

T. Fisher, Hyperbolic sets with nonempty interior,, Discrete Contin. Dyn. Syst., 15 (2006), 433. doi: doi:10.3934/dcds.2006.15.433.

[6]

J. Franks, Anosov diffeomorphisms,, in, (1970), 61.

[7]

V. Z. Grines, V. S. Medvedev, and E. V. Zhuzhoma, On surface attractors and repellers in 3-manifolds,, Mat. Zametki, 78 (2005), 813.

[8]

B. Günther, Attractors which are homeomorphic to compact abelian groups,, Manuscripta Math., 82 (1994), 31. doi: doi:10.1007/BF02567683.

[9]

K. Hiraide, A simple proof of the Franks-Newhouse theorem on codimension-one Anosov diffeomorphisms,, Ergodic Theory Dynam. Systems, 21 (2001), 801. doi: doi:10.1017/S0143385701001390.

[10]

W. Hurewicz and H. Wallman, "Dimension Theory,", Princeton University Press, (1941).

[11]

B. Jiang, S. Wang, and H. Zheng, No embeddings of solenoids into surfaces,, Proc. Amer. Math. Soc., 136 (2008), 3697. doi: doi:10.1090/S0002-9939-08-09340-4.

[12]

J. L. Kaplan, J. Mallet-Paret, and J. A. Yorke, The Lyapunov dimension of a nowhere differentiable attracting torus,, Ergodic Theory Dynam. Systems, 4 (1984), 261. doi: doi:10.1017/S0143385700002431.

[13]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Cambridge University Press, (1995).

[14]

A. Manning, There are no new Anosov diffeomorphisms on tori,, Amer. J. Math., 96 (1974), 422. doi: doi:10.2307/2373551.

[15]

S. E. Newhouse, On codimension one Anosov diffeomorphisms,, Amer. J. Math., 92 (1970), 761. doi: doi:10.2307/2373372.

[16]

R. V. Plykin, The topology of basic sets of Smale diffeomorphisms,, Math. USSR-Sb., 13 (1971), 297. doi: doi:10.1070/SM1971v013n02ABEH001026.

[17]

R. V. Plykin, Hyperbolic attractors of diffeomorphisms,, Russian Math. Surveys, 35 (1980), 109. doi: doi:10.1070/RM1980v035n03ABEH001702.

[18]

R. V. Plykin, Hyperbolic attractors of diffeomorphisms (the nonorientable case),, Russian Math. Surveys, 35 (1980), 186. doi: doi:10.1070/RM1980v035n04ABEH001879.

[19]

D. Ruelle and D. Sullivan, Currents, flows and diffeomorphisms,, Topology, 14 (1975), 319. doi: doi:10.1016/0040-9383(75)90016-6.

[20]

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

[21]

R. F. Williams, One-dimensional non-wandering sets,, Topology, 6 (1967), 473. doi: doi:10.1016/0040-9383(67)90005-5.

[22]

R. F. Williams, Classification of one dimensional attractors,, in, (1970), 341.

[23]

R. F. Williams, Expanding attractors,, Inst. Hautes Études Sci. Publ. Math., (1974), 169.

show all references

References:
[1]

C. Bonatti, Problem in dynamical systems,, , (1999).

[2]

H. G. Bothe, Expanding attractors with stable foliations of class $C^0$,, in, 1514 (1992), 36.

[3]

B. Brenken, The local product structure of expansive automorphisms of solenoids and their associated $C^$*-algebras,, Canad. J. Math., 48 (1996), 692.

[4]

A. Brown, Constraints on dynamics preserving certain hyperbolic sets,, Ergodic Theory Dynam. Systems, ().

[5]

T. Fisher, Hyperbolic sets with nonempty interior,, Discrete Contin. Dyn. Syst., 15 (2006), 433. doi: doi:10.3934/dcds.2006.15.433.

[6]

J. Franks, Anosov diffeomorphisms,, in, (1970), 61.

[7]

V. Z. Grines, V. S. Medvedev, and E. V. Zhuzhoma, On surface attractors and repellers in 3-manifolds,, Mat. Zametki, 78 (2005), 813.

[8]

B. Günther, Attractors which are homeomorphic to compact abelian groups,, Manuscripta Math., 82 (1994), 31. doi: doi:10.1007/BF02567683.

[9]

K. Hiraide, A simple proof of the Franks-Newhouse theorem on codimension-one Anosov diffeomorphisms,, Ergodic Theory Dynam. Systems, 21 (2001), 801. doi: doi:10.1017/S0143385701001390.

[10]

W. Hurewicz and H. Wallman, "Dimension Theory,", Princeton University Press, (1941).

[11]

B. Jiang, S. Wang, and H. Zheng, No embeddings of solenoids into surfaces,, Proc. Amer. Math. Soc., 136 (2008), 3697. doi: doi:10.1090/S0002-9939-08-09340-4.

[12]

J. L. Kaplan, J. Mallet-Paret, and J. A. Yorke, The Lyapunov dimension of a nowhere differentiable attracting torus,, Ergodic Theory Dynam. Systems, 4 (1984), 261. doi: doi:10.1017/S0143385700002431.

[13]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Cambridge University Press, (1995).

[14]

A. Manning, There are no new Anosov diffeomorphisms on tori,, Amer. J. Math., 96 (1974), 422. doi: doi:10.2307/2373551.

[15]

S. E. Newhouse, On codimension one Anosov diffeomorphisms,, Amer. J. Math., 92 (1970), 761. doi: doi:10.2307/2373372.

[16]

R. V. Plykin, The topology of basic sets of Smale diffeomorphisms,, Math. USSR-Sb., 13 (1971), 297. doi: doi:10.1070/SM1971v013n02ABEH001026.

[17]

R. V. Plykin, Hyperbolic attractors of diffeomorphisms,, Russian Math. Surveys, 35 (1980), 109. doi: doi:10.1070/RM1980v035n03ABEH001702.

[18]

R. V. Plykin, Hyperbolic attractors of diffeomorphisms (the nonorientable case),, Russian Math. Surveys, 35 (1980), 186. doi: doi:10.1070/RM1980v035n04ABEH001879.

[19]

D. Ruelle and D. Sullivan, Currents, flows and diffeomorphisms,, Topology, 14 (1975), 319. doi: doi:10.1016/0040-9383(75)90016-6.

[20]

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

[21]

R. F. Williams, One-dimensional non-wandering sets,, Topology, 6 (1967), 473. doi: doi:10.1016/0040-9383(67)90005-5.

[22]

R. F. Williams, Classification of one dimensional attractors,, in, (1970), 341.

[23]

R. F. Williams, Expanding attractors,, Inst. Hautes Études Sci. Publ. Math., (1974), 169.

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