On the subriemannian geometry of contact Anosov flows

TL;DR

This paper explores the relationship between subriemannian geometry and contact Anosov flows, revealing how geodesic energy is distributed between stable and unstable directions, with explicit solutions involving harmonic oscillators.

Contribution

It establishes a novel link between subriemannian geodesics and hyperbolic dynamics, providing explicit energy-sharing results for specific contact Anosov flows and extending to higher dimensions.

Findings

Energy of subriemannian geodesics is equally shared between stable and unstable projections in certain contact Anosov flows.

Explicit solutions involve Jacobi elliptic functions and harmonic oscillator equations.

Results extend to geodesic flows on negatively curved Riemannian manifolds.

Abstract

We investigate certain natural connections between subriemannian geometry and hyperbolic dynamical systems. In particular, we study dynamically defined horizontal distributions which split into two integrable ones and ask: how is the energy of a subriemannian geodesic shared between its projections onto the integrable summands? We show that if the horizontal distribution is the sum of the strong stable and strong unstable distributions of a special type of a contact Anosov flow in three dimensions, then for any short enough subriemannian geodesic connecting points on the same orbit of the Anosov flow, the energy of the geodesic is shared \emph{equally} between its projections onto the stable and unstable bundles. The proof relies on a connection between the geodesic equations and the harmonic oscillator equation, and its explicit solution by the Jacobi elliptic functions. Using a different idea, we prove an analogous result in higher dimensions for the geodesic flow of a closed Riemannian manifold of constant negative curvature.