POSITIVE-ENTROPY HAMILTONIAN SYSTEMS ON NILMANIFOLDS VIA SCATTERING

Abstract. 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 t4 of T4 has the standard basis {Xij} 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 Xi,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 TΣ has positive topological entropy and is real-analytically non-integrable. It extends earlier work by Butler and Gelfreich. 1