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Integrable geodesic flows with wild first integrals: the case of two-step nilmanifolds

Published online by Cambridge University Press:  20 June 2003

LEO BUTLER
LEO BUTLER
Affiliation:
Department of Mathematics Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA (e-mail: [email protected])
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Abstract

This paper has four main results: (i) it shows that left-invariant geodesic flows on a broad class of two-step nilmanifolds—which are dubbed almost non-singular—are integrable in the non-commutative sense of Nehoros˘ev; (ii) the left-invariant geodesic flows on all Heisenberg–Reiter nilmanifolds are Liouville integrable; (iii) the topological entropy of a left-invariant geodesic flow on a two-step nilmanifold vanishes; (iv) there exist two-step nilmanifolds with non-integrable left-invariant geodesic flows. It is also shown that for each of the integrable Hamiltonians investigated here, there is a C^2-open neighbourhood in C^2(T^* M) such that every integrable Hamiltonian vector field in this neighbourhood must have wild first integrals.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 23 , Issue 3 , June 2003 , pp. 771 - 797
Copyright
2003 Cambridge University Press

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