Positiveentropy Hamiltonian systems on Nilmanifolds via scattering
Abstract
Let Σ be a compact quotient of T_{4}, the Lie group of 4 × 4 upper triangular matrices with unity along the diagonal. The Lie algebra {\mathfrak 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 subRiemannian 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 nonintegrable. This paper proves that the geodesic flow of that Carnot metric on TΣ has positive topological entropy and its Euler field is realanalytically nonintegrable. It extends earlier work by Butler and Gelfreich.
 Publication:

Nonlinearity
 Pub Date:
 October 2014
 DOI:
 10.1088/09517715/27/10/2479
 arXiv:
 arXiv:1402.2122
 Bibcode:
 2014Nonli..27.2479B
 Keywords:

 Nonlinear Sciences  Chaotic Dynamics;
 Mathematical Physics;
 37J30;
 53C17;
 53C30;
 53D25
 EPrint:
 3 figures