Positive-entropy Hamiltonian systems on Nilmanifolds via scattering

Let Σ be a compact quotient of T₄, the Lie group of 4 × 4 upper triangular matrices with unity along the diagonal. The Lie algebra {\mathfrak t}₄ of T₄ 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 T₄ for which X_{i, 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 TΣ has positive topological entropy and its Euler field is real-analytically non-integrable. It extends earlier work by Butler and Gelfreich.