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Nonexpanding attractors: Conjugacy to algebraic models and classification in 3-manifolds
1. | Department of Mathematics, Tufts University, Medford, MA 02155, United States |
References:
[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. Brown, Constraints 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. |
show all references
References:
[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. Brown, Constraints 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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