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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 - A, 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.

[2]

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

[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. doi: 10.3934/jmd.2009.3.1.

[4]

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

[5]

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

[6]

A. Hammerlindl, "Leaf Conjugacies on the Torus,", Ph.D thesis, (2009).

[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. doi: 10.3934/jmd.2008.2.187.

[8]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,", 583 of Lecture Notes in Mathematics, 583 (1977).

[9]

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

[10]

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

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.

[2]

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

[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. doi: 10.3934/jmd.2009.3.1.

[4]

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

[5]

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

[6]

A. Hammerlindl, "Leaf Conjugacies on the Torus,", Ph.D thesis, (2009).

[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. doi: 10.3934/jmd.2008.2.187.

[8]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,", 583 of Lecture Notes in Mathematics, 583 (1977).

[9]

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

[10]

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

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