Positve Entropy Geodesic Flows on Nilmanifolds

TL;DR

This paper demonstrates the existence of horseshoes in the Euler equations for specific metrics on a 4x4 nilpotent group, indicating chaotic dynamics and non-integrability.

Contribution

It establishes the presence of horseshoes in Euler equations for certain metrics on the nilpotent group T, combining theoretical and numerical methods.

Findings

Horseshoes exist in Euler equations for specific metrics on T

Numerical Melnikov integral supports chaotic behavior

Extends non-integrability results of previous studies

Abstract

Let T be the nilpotent group of 4 x 4 real upper triangular matrices. In this note we show that the Euler equations of certain left-invariant riemannian metrics on T have a horseshoe. We also show, with the aid of a numerical computation of a Melnikov-type integral, that the Euler equations of the sub-riemannian Carnot metric on T has a horseshoe. This sharpens an earlier result of Montgomery, Shapiro and Stolin who had shown that the equations are algebraically non-integrable.