American Institute of Mathematical Sciences

July  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, http://www.math.sunysb.edu/dynamics/bonatti_prob.txt, November 1999. Google Scholar [2] H. G. Bothe, Expanding attractors with stable foliations of class $C^0$, in "Ergodic theory and related topics, III," Lecture Notes in Math., 1514, Springer, Berlin, (1992), 36-61.  Google Scholar [3] B. Brenken, The local product structure of expansive automorphisms of solenoids and their associated $C^$*-algebras, Canad. J. Math., 48 (1996), 692-709.  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), 433-446. doi: doi:10.3934/dcds.2006.15.433.  Google Scholar [6] J. Franks, Anosov diffeomorphisms, in "Global Analysis," Amer. Math. Soc., Providence, R.I., 1970, 61-93.  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), 813-826.  Google Scholar [8] B. Günther, Attractors which are homeomorphic to compact abelian groups, Manuscripta Math., 82 (1994), 31-40. 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), 801-806. doi: doi:10.1017/S0143385701001390.  Google Scholar [10] W. Hurewicz and H. Wallman, "Dimension Theory," Princeton University Press, Princeton, N. J., 1941.  Google Scholar [11] B. Jiang, S. Wang, and H. Zheng, No embeddings of solenoids into surfaces, Proc. Amer. Math. Soc., 136 (2008), 3697-3700. 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), 261-281. doi: doi:10.1017/S0143385700002431.  Google Scholar [13] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, Cambridge, 1995.  Google Scholar [14] A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422-429. doi: doi:10.2307/2373551.  Google Scholar [15] S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 (1970), 761-770. doi: doi:10.2307/2373372.  Google Scholar [16] R. V. Plykin, The topology of basic sets of Smale diffeomorphisms, Math. USSR-Sb., 13 (1971), 297-307. doi: doi:10.1070/SM1971v013n02ABEH001026.  Google Scholar [17] R. V. Plykin, Hyperbolic attractors of diffeomorphisms, Russian Math. Surveys, 35 (1980), 109-121. doi: doi:10.1070/RM1980v035n03ABEH001702.  Google Scholar [18] R. V. Plykin, Hyperbolic attractors of diffeomorphisms (the nonorientable case), Russian Math. Surveys, 35 (1980), 186-187. doi: doi:10.1070/RM1980v035n04ABEH001879.  Google Scholar [19] D. Ruelle and D. Sullivan, Currents, flows and diffeomorphisms, Topology, 14 (1975), 319-327. doi: doi:10.1016/0040-9383(75)90016-6.  Google Scholar [20] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: doi:10.1090/S0002-9904-1967-11798-1.  Google Scholar [21] R. F. Williams, One-dimensional non-wandering sets, Topology, 6 (1967), 473-487. doi: doi:10.1016/0040-9383(67)90005-5.  Google Scholar [22] R. F. Williams, Classification of one dimensional attractors, in "Global Analysis," Amer. Math. Soc., Providence, R.I., 1970, 341-361.  Google Scholar [23] R. F. Williams, Expanding attractors, Inst. Hautes Études Sci. Publ. Math., (1974), 169-203.  Google Scholar

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References:
 [1] C. Bonatti, Problem in dynamical systems, http://www.math.sunysb.edu/dynamics/bonatti_prob.txt, November 1999. Google Scholar [2] H. G. Bothe, Expanding attractors with stable foliations of class $C^0$, in "Ergodic theory and related topics, III," Lecture Notes in Math., 1514, Springer, Berlin, (1992), 36-61.  Google Scholar [3] B. Brenken, The local product structure of expansive automorphisms of solenoids and their associated $C^$*-algebras, Canad. J. Math., 48 (1996), 692-709.  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), 433-446. doi: doi:10.3934/dcds.2006.15.433.  Google Scholar [6] J. Franks, Anosov diffeomorphisms, in "Global Analysis," Amer. Math. Soc., Providence, R.I., 1970, 61-93.  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), 813-826.  Google Scholar [8] B. Günther, Attractors which are homeomorphic to compact abelian groups, Manuscripta Math., 82 (1994), 31-40. 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), 801-806. doi: doi:10.1017/S0143385701001390.  Google Scholar [10] W. Hurewicz and H. Wallman, "Dimension Theory," Princeton University Press, Princeton, N. J., 1941.  Google Scholar [11] B. Jiang, S. Wang, and H. Zheng, No embeddings of solenoids into surfaces, Proc. Amer. Math. Soc., 136 (2008), 3697-3700. 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), 261-281. doi: doi:10.1017/S0143385700002431.  Google Scholar [13] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, Cambridge, 1995.  Google Scholar [14] A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422-429. doi: doi:10.2307/2373551.  Google Scholar [15] S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 (1970), 761-770. doi: doi:10.2307/2373372.  Google Scholar [16] R. V. Plykin, The topology of basic sets of Smale diffeomorphisms, Math. USSR-Sb., 13 (1971), 297-307. doi: doi:10.1070/SM1971v013n02ABEH001026.  Google Scholar [17] R. V. Plykin, Hyperbolic attractors of diffeomorphisms, Russian Math. Surveys, 35 (1980), 109-121. doi: doi:10.1070/RM1980v035n03ABEH001702.  Google Scholar [18] R. V. Plykin, Hyperbolic attractors of diffeomorphisms (the nonorientable case), Russian Math. Surveys, 35 (1980), 186-187. doi: doi:10.1070/RM1980v035n04ABEH001879.  Google Scholar [19] D. Ruelle and D. Sullivan, Currents, flows and diffeomorphisms, Topology, 14 (1975), 319-327. doi: doi:10.1016/0040-9383(75)90016-6.  Google Scholar [20] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: doi:10.1090/S0002-9904-1967-11798-1.  Google Scholar [21] R. F. Williams, One-dimensional non-wandering sets, Topology, 6 (1967), 473-487. doi: doi:10.1016/0040-9383(67)90005-5.  Google Scholar [22] R. F. Williams, Classification of one dimensional attractors, in "Global Analysis," Amer. Math. Soc., Providence, R.I., 1970, 341-361.  Google Scholar [23] R. F. Williams, Expanding attractors, Inst. Hautes Études Sci. Publ. Math., (1974), 169-203.  Google Scholar
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