Positive-entropy Hamiltonian systems on Nilmanifolds via scattering

Let Sigma be a compact quotient of T-4, the Lie group of 4 x 4 upper triangular matrices with unity along the diagonal. The Lie algebra t(4) of T-4 has the standard basis {X-ij} of matrices with 0 everywhere but in the (i, j) entry, which is unity. Let g be the Carnot metric, a sub-Riemannian metric, on T-4 for which X-i,X- i+1, (i = 1, 2, 3), is an orthonormal basis. Montgomery, Shapiro and Stolin showed that the geodesic flow of g is algebraically non-integrable. This paper proves that the geodesic flow of that Carnot metric on T Sigma has positive topological entropy and its Euler field is real-analytically non-integrable. It extends earlier work by Butler and Gelfreich.