Zero entropy, non-integrable geodesic flows and a non-commutative rotation vector

Abstract:

Let $\mathfrak g$ be a $2$-step nilpotent Lie algebra; we say $\mathfrak g$ is non-integrable if, for a generic pair of points $p,p’ \in \mathfrak g^*$, the isotropy algebras do not commute: $[\mathfrak g_p,\mathfrak g_{p’}] \neq 0$. Theorem: If $G$ is a simply-connected $2$-step nilpotent Lie group, $\mathfrak g ={\mathrm {Lie}}(G)$ is non-integrable, $D < G$ is a cocompact subgroup, and ${\mathbf g}$ is a left-invariant Riemannian metric, then the geodesic flow of ${\mathbf g}$ on $T^*(D \backslash G)$ is neither Liouville nor non-commutatively integrable with $C^0$ first integrals. The proof uses a generalization of the rotation vector pioneered by Benardete and Mitchell.