American Institute of Mathematical Sciences

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

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

Aaron W. Brown 1,

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].
Keywords: classification., conjugacy, solenoids, Hyperbolic attractors.
Mathematics Subject Classification: Primary: 37C70, 37C15; Secondary: 37D2.
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). Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[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. Google Scholar

[10]

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

[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. Google Scholar

[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. Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[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. Google Scholar

[10]

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

[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. Google Scholar

[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. Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[1]

Ronald de Man. On composants of solenoids. Electronic Research Announcements, 1995, 1: 87-90.

[2]

B. San Martín, Kendry J. Vivas. Asymptotically sectional-hyperbolic attractors. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4057-4071. doi: 10.3934/dcds.2019163
[3]

Vladimir Georgiev, Eugene Stepanov. Metric cycles, curves and solenoids. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1443-1463. doi: 10.3934/dcds.2014.34.1443
[4]

A. M. López. Finiteness and existence of attractors and repellers on sectional hyperbolic sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 337-354. doi: 10.3934/dcds.2017014
[5]

Aubin Arroyo, Enrique R. Pujals. Dynamical properties of singular-hyperbolic attractors. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 67-87. doi: 10.3934/dcds.2007.19.67
[6]

V. V. Chepyzhov, A. Miranville. Trajectory and global attractors of dissipative hyperbolic equations with memory. Communications on Pure & Applied Analysis, 2005, 4 (1) : 115-142. doi: 10.3934/cpaa.2005.4.115
[7]

Peidong Liu, Kening Lu. A note on partially hyperbolic attractors: Entropy conjecture and SRB measures. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 341-352. doi: 10.3934/dcds.2015.35.341
[8]

Keith Burns, Dmitry Dolgopyat, Yakov Pesin, Mark Pollicott. Stable ergodicity for partially hyperbolic attractors with negative central exponents. Journal of Modern Dynamics, 2008, 2 (1) : 63-81. doi: 10.3934/jmd.2008.2.63
[9]

Zhicong Liu. SRB attractors with intermingled basins for non-hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1545-1562. doi: 10.3934/dcds.2013.33.1545
[10]

Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 215-234. doi: 10.3934/dcds.2008.22.215
[11]

Carlos H. Vásquez. Stable ergodicity for partially hyperbolic attractors with positive central Lyapunov exponents. Journal of Modern Dynamics, 2009, 3 (2) : 233-251. doi: 10.3934/jmd.2009.3.233
[12]

Zeng Lian, Peidong Liu, Kening Lu. Existence of SRB measures for a class of partially hyperbolic attractors in banach spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3905-3920. doi: 10.3934/dcds.2017164
[13]

Manfred Einsiedler and Elon Lindenstrauss. Rigidity properties of \zd-actions on tori and solenoids. Electronic Research Announcements, 2003, 9: 99-110.

[14]

Francisco J. López-Hernández. Dynamics of induced homeomorphisms of one-dimensional solenoids. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4243-4257. doi: 10.3934/dcds.2018185
[15]

A. Yu. Ol'shanskii and M. V. Sapir. The conjugacy problem for groups, and Higman embeddings. Electronic Research Announcements, 2003, 9: 40-50.

[16]

Shan Ma, Chengkui Zhong. The attractors for weakly damped non-autonomous hyperbolic equations with a new class of external forces. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 53-70. doi: 10.3934/dcds.2007.18.53
[17]

Solange Mancini, Miriam Manoel, Marco Antonio Teixeira. Divergent diagrams of folds and simultaneous conjugacy of involutions. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 657-674. doi: 10.3934/dcds.2005.12.657
[18]

Kingshook Biswas. Complete conjugacy invariants of nonlinearizable holomorphic dynamics. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 847-856. doi: 10.3934/dcds.2010.26.847
[19]

Yong Fang. On smooth conjugacy of expanding maps in higher dimensions. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 687-697. doi: 10.3934/dcds.2011.30.687
[20]

Adriano Da Silva, Alexandre J. Santana, Simão N. Stelmastchuk. Topological conjugacy of linear systems on Lie groups. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3411-3421. doi: 10.3934/dcds.2017144

2018 Impact Factor: 0.295

Tools

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences