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

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  • 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].
    Mathematics Subject Classification: Primary: 37C70, 37C15; Secondary: 37D20.

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  • [1]

    C. Bonatti, Problem in dynamical systems, http://www.math.sunysb.edu/dynamics/bonatti_prob.txt, November 1999.

    [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.

    [3]

    B. Brenken, The local product structure of expansive automorphisms of solenoids and their associated $C^$*-algebras, Canad. J. Math., 48 (1996), 692-709.

    [4]

    A. BrownConstraints on dynamics preserving certain hyperbolic sets, Ergodic Theory Dynam. Systems, to appear.

    [5]

    T. Fisher, Hyperbolic sets with nonempty interior, Discrete Contin. Dyn. Syst., 15 (2006), 433-446.doi: doi:10.3934/dcds.2006.15.433.

    [6]

    J. Franks, Anosov diffeomorphisms, in "Global Analysis," Amer. Math. Soc., Providence, R.I., 1970, 61-93.

    [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.

    [8]

    B. Günther, Attractors which are homeomorphic to compact abelian groups, Manuscripta Math., 82 (1994), 31-40.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), 801-806.doi: doi:10.1017/S0143385701001390.

    [10]

    W. Hurewicz and H. Wallman, "Dimension Theory," Princeton University Press, Princeton, N. J., 1941.

    [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.

    [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.

    [13]

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

    [14]

    A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422-429.doi: doi:10.2307/2373551.

    [15]

    S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 (1970), 761-770.doi: doi:10.2307/2373372.

    [16]

    R. V. Plykin, The topology of basic sets of Smale diffeomorphisms, Math. USSR-Sb., 13 (1971), 297-307.doi: doi:10.1070/SM1971v013n02ABEH001026.

    [17]

    R. V. Plykin, Hyperbolic attractors of diffeomorphisms, Russian Math. Surveys, 35 (1980), 109-121.doi: doi:10.1070/RM1980v035n03ABEH001702.

    [18]

    R. V. Plykin, Hyperbolic attractors of diffeomorphisms (the nonorientable case), Russian Math. Surveys, 35 (1980), 186-187.doi: doi:10.1070/RM1980v035n04ABEH001879.

    [19]

    D. Ruelle and D. Sullivan, Currents, flows and diffeomorphisms, Topology, 14 (1975), 319-327.doi: doi:10.1016/0040-9383(75)90016-6.

    [20]

    S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.doi: doi:10.1090/S0002-9904-1967-11798-1.

    [21]

    R. F. Williams, One-dimensional non-wandering sets, Topology, 6 (1967), 473-487.doi: doi:10.1016/0040-9383(67)90005-5.

    [22]

    R. F. Williams, Classification of one dimensional attractors, in "Global Analysis," Amer. Math. Soc., Providence, R.I., 1970, 341-361.

    [23]

    R. F. Williams, Expanding attractors, Inst. Hautes Études Sci. Publ. Math., (1974), 169-203.

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