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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
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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  • 1. H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. (3) 25, 603-614 (1972). MR 52:577
  • 2. D. Benardete and J. Mitchell, Asymptotic homotopy cycles for flows and $\Pi \sb 1$ de Rham theory, Trans. Amer. Math. Soc. 338(2), 495-535 (1993). MR 93j:58107
  • 3. A. V. Bolsinov and I. A. Taimanov, Integrable geodesic flows with positive topological entropy, Invent. Math. 140(3), 639-650 (2000). MR 2001b:37081
  • 4. L. T. Butler, A new class of homogeneous manifolds with Liouville-integrable geodesic flows, C. R. Math. Acad. Sci. Soc. R. Can. 21(4), 127-131 (1999). MR 2001i:53141
  • 5. L. T. Butler, New examples of integrable geodesic flows, Asian J. Math. 4(3), 515-526 (2000). MR 2001i:37090
  • 6. L. T. Butler, Integrable geodesic flows with wild first integrals: The case of two-step nilmanifolds, Ergodic Theory Dynam. Systems, to appear.
  • 7. L. T. Butler, Integrable geodesic flows on $n$-step nilmanifolds, J. Geom. Phys. 36(3-4), 315-323 (2000). MR 2002j:37077
  • 8. L. T. Butler, Invariant metrics on a nilmanifold with positive topological entropy, submitted to Geometriae Dedicata. 2001.
  • 9. K. T. Chen, Extension of ${C}\sp{\infty }$ function algebra by integrals and Malcev completion of $\pi \sb{1}$. Advances in Math. 23(2), 181-210 (1977). MR 56:16664
  • 10. P. Eberlein, Geometry of $2$-step nilpotent groups with a left invariant metric, Ann. Sci. École Norm. Sup. 27(5), 611-660 (1994). MR 95m:53059
  • 11. P. Eberlein, Geometry of $2$-step nilpotent groups with a left invariant metric. II, Trans. Amer. Math. Soc. 343(2), 805-828 (1994). MR 95b:53061
  • 12. F. Fried, The geometry of cross sections to flows, Topology 21(4), 353-371 (1982). MR 84d:58068
  • 13. C. S. Gordon and E. N. Wilson, The spectrum of the Laplacian on Riemannian Heisenberg manifolds, Michigan Math. J. 33(2), 253-271 (1986). MR 87k:58275
  • 14. A. B. Katok, Ergodic perturbations of degenerate integrable Hamiltonian systems, Izv. Akad. Nauk SSSR Ser. Mat. 37, 539-576 (1973). MR 48:9758
  • 15. K. B. Lee and K. Park, Smoothly closed geodesics in $2$-step nilmanifolds, Indiana Univ. Math. J. 45(1), 1-14 (1996). MR 97h:53044
  • 16. A. I. Malcev, On a class of homogeneous spaces, Amer. Math. Soc. Translation, no. 39, 1951. MR 12:589e
  • 17. M. Mast, Closed geodesics in 2-step nilmanifolds, Indiana Univ. Math. J. 43, 885-911 (1994). MR 96a:53057
  • 18. G. P. Paternain, Geodesic flows, Progress in Math., vol. 180, Birkhäuser, Boston, MA, 1999. MR 2000h:53108
  • 19. F. Rhodes, Asymptotic cycles for continuous curves on geodesic spaces, J. London Math. Soc. (2), 6, 247-255 (1973). MR 49:6208
  • 20. S. Schwartzman, Asymptotic cycles, Ann. of Math. (2) 66, 270-284 (1957). MR 19:568i

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Additional Information

Leo T. Butler
Affiliation: Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208

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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