A lower bound on the subriemannian distance for Hölder distributions

Abstract:

Whereas subriemannian geometry usually deals with smooth horizontal distributions, partially hyperbolic dynamical systems provide many examples of subriemannian geometries defined by non-smooth (namely, Hölder continuous) distributions. These distributions are of great significance for the behavior of the parent dynamical system. The study of Hölder subriemannian geometries could therefore offer new insights into both dynamics and subriemannian geometry. In this paper we make a small step in that direction: we prove a Hölder-type lower bound on the subriemannian distance for Hölder continuous nowhere integrable codimension one distributions. This bound generalizes the well-known square root bound valid in the smooth case.