An example of an integrable geodesic flow with positive entropy
by
Alexey V. Bolsinov
Moscow State University
Coauthors: Iskander A. Taimanov (Novosibirsk Institute of Mathematics)

The main result of this work is the following theorem.

There is a real-analytic Riemannian manifold M_A diffeomorphic to the quotient of T² ×R¹ with respect to a free Z-action generated by the map

(X, z) --> (AX, z+1),

where X = (x, y) in T² = R²/Z², z in R, and A is an Anosov automorphism of the 2-torus T² defined by the matrix

A = æ ç è 2 1 1 1 ö ÷ ø ,

(1)

such that

i) the geodesic flow on M_A is (Liouville) integrable by C^\infty first integrals;

ii) the geodesic flow on M_A is not (Liouville) integrable by real-analytic first integrals;

iii) the topological entropy of the geodesic flow F_t is positive;

iv) the fundamental group \pi₁(M_A) of the manifold M_A has exponential growth;

v) the unit covector bundle S M_A contains a submanifold N such that N is diffeomorphic to the 2-torus T² and the restriction of F₁ onto N is an Anosov automorphism given by matrix (1).

Date received: June 2, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacy-31.