Positive-entropy Hamiltonian systems on Nilmanifolds via Scattering

Let $\Sigma$ be a compact quotient of $T_4$, the Lie group of $4 \times 4$ 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 $X_{i,i+1}$, $(i=1,2,3)$, is an orthonormal basis. Montgomery, Shapiro and Stolin showed that the geodesic flow of $g$ is algebraically non-integrable. This note proves that the geodesic flow of that Carnot metric on $T \Sigma$ has positive topological entropy and is real-analytically non-integrable. It extends earlier work by Butler and Gelfreich.