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August  2013, 33(8): 3641-3669. doi: 10.3934/dcds.2013.33.3641

Partial hyperbolicity on 3-dimensional nilmanifolds

Andy Hammerlindl 1,

1. 

School of Mathematics and Statistics, University of Sydney, NSW, 2006, Australia

Received  August 2012 Revised  November 2012 Published  January 2013

Every partially hyperbolic diffeomorphism on a 3-dimensional nilmanifold is leaf conjugate to a nilmanifold automorphism. Moreover, if the nilmanifold is not the 3-torus, the center foliation is an invariant circle bundle.
Keywords: Partial hyperbolicity, classification..
Mathematics Subject Classification: Primary: 37D3.
Citation: Andy Hammerlindl. Partial hyperbolicity on 3-dimensional nilmanifolds. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3641-3669. doi: 10.3934/dcds.2013.33.3641
References:
[1]

L. Auslander, Bieberbach's theorems on space groups and discrete uniform subgroups of Lie groups, Annals of Math., 71 (1960), 579-590.  Google Scholar

[2]

M. Brin, On dynamical coherence, Ergod. Th. and Dynam. Sys., 23 (2003), 395-401. doi: 10.1017/S0143385702001499.  Google Scholar

[3]

M. Brin, D. Burago and S. Ivanov, Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus, Journal of Modern Dynamics, 3 (2009), 1-11. doi: 10.3934/jmd.2009.3.1.  Google Scholar

[4]

J. Franks, Anosov diffeomorphisms on tori, Transactions of the American Mathematical Society, 145 (1969), 117-124.  Google Scholar

[5]

J. Franks, Anosov diffeomorphisms, Global Analysis: Proceedings of the Symposia in Pure Mathematics, 14 (1970), 61-93.  Google Scholar

[6]

A. Hammerlindl, "Leaf Conjugacies on the Torus," Ph.D thesis, University of Toronto, 2009.  Google Scholar

[7]

F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, Partial hyperbolicity and ergodicity in dimension three, Journal of Modern Dynamics, 2 (2008), 187-208. doi: 10.3934/jmd.2008.2.187.  Google Scholar

[8]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," 583 of Lecture Notes in Mathematics, Springer-Verlag, 1977.  Google Scholar

[9]

A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422-429.  Google Scholar

[10]

K. Parwani, On 3-manifolds that support partially hyperbolic diffeomorphisms, Nonlinearity, 23 (2010), 589-606. doi: 10.1088/0951-7715/23/3/009.  Google Scholar

show all references

References:
[1]

L. Auslander, Bieberbach's theorems on space groups and discrete uniform subgroups of Lie groups, Annals of Math., 71 (1960), 579-590.  Google Scholar

[2]

M. Brin, On dynamical coherence, Ergod. Th. and Dynam. Sys., 23 (2003), 395-401. doi: 10.1017/S0143385702001499.  Google Scholar

[3]

M. Brin, D. Burago and S. Ivanov, Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus, Journal of Modern Dynamics, 3 (2009), 1-11. doi: 10.3934/jmd.2009.3.1.  Google Scholar

[4]

J. Franks, Anosov diffeomorphisms on tori, Transactions of the American Mathematical Society, 145 (1969), 117-124.  Google Scholar

[5]

J. Franks, Anosov diffeomorphisms, Global Analysis: Proceedings of the Symposia in Pure Mathematics, 14 (1970), 61-93.  Google Scholar

[6]

A. Hammerlindl, "Leaf Conjugacies on the Torus," Ph.D thesis, University of Toronto, 2009.  Google Scholar

[7]

F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, Partial hyperbolicity and ergodicity in dimension three, Journal of Modern Dynamics, 2 (2008), 187-208. doi: 10.3934/jmd.2008.2.187.  Google Scholar

[8]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," 583 of Lecture Notes in Mathematics, Springer-Verlag, 1977.  Google Scholar

[9]

A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422-429.  Google Scholar

[10]

K. Parwani, On 3-manifolds that support partially hyperbolic diffeomorphisms, Nonlinearity, 23 (2010), 589-606. doi: 10.1088/0951-7715/23/3/009.  Google Scholar

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