A lower bound on the subriemannian distance for H\"older distributions

TL;DR

This paper establishes a H"older-type lower bound on subriemannian distance for non-smooth, H"older continuous distributions, extending classical results from smooth to non-smooth geometries relevant in dynamical systems.

Contribution

It provides the first known lower bound for subriemannian distance in H"older continuous, nowhere integrable distributions, generalizing the smooth case square root bound.

Findings

Proves a H"older-type lower bound on subriemannian distance.

Extends classical bounds from smooth to H"older continuous distributions.

Highlights relevance to partially hyperbolic dynamical systems.

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, H\"older continuous) distributions. These distributions are of great significance for the behavior of the parent dynamical system. The study of H\"older 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 H\"older-type lower bound on the subriemannian distance for H\"older continuous nowhere integrable codimension one distributions. This bound generalizes the well-known square root bound valid in the smooth case.