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

Abstract

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