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Zero entropy, non-integrable geodesic flows and a non-commutative rotation vector


Author: Leo T. Butler
Journal: Trans. Amer. Math. Soc. 355 (2003), 3641-3650
MSC (2000): Primary 37J30, 37E45; Secondary 53D25
DOI: https://doi.org/10.1090/S0002-9947-03-03334-8
Published electronically: May 15, 2003
MathSciNet review: 1990166
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Abstract: Let $\mathfrak g$ be a $2$-step nilpotent Lie algebra; we say $\mathfrak g$ is non-integrable if, for a generic pair of points $p,p' \in \mathfrak g^*$, the isotropy algebras do not commute: $[\mathfrak g_p,\mathfrak g_{p'}] \neq 0$. Theorem: If $G$ is a simply-connected $2$-step nilpotent Lie group, $\mathfrak g ={\mathrm {Lie}}(G) $ is non-integrable, $D < G$ is a cocompact subgroup, and ${\mathbf g}$ is a left-invariant Riemannian metric, then the geodesic flow of ${\mathbf g}$ on $T^*(D \backslash G)$ is neither Liouville nor non-commutatively integrable with $C^0$ first integrals. The proof uses a generalization of the rotation vector pioneered by Benardete and Mitchell.


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Additional Information

Leo T. Butler
Affiliation: Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208
Email: lbutler@math.northwestern.edu

DOI: https://doi.org/10.1090/S0002-9947-03-03334-8
Keywords: Rotation\, vector, geodesic\, flows, entropy, nilmanifolds, nonintegrability
Received by editor(s): May 13, 2002
Received by editor(s) in revised form: September 23, 2002
Published electronically: May 15, 2003
Additional Notes: Research partially supported by a Natural Sciences and Engineering Research Council of Canada Postdoctoral Fellowship. Thanks to Gabriel Paternain, John Franks and Queen’s University.
Article copyright: © Copyright 2003 American Mathematical Society

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