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

## Partial hyperbolicity on 3-dimensional nilmanifolds

 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.
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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