On The Limiting Distribution of Free Path Lengths for Flat Surfaces with Circular Obstacles

TL;DR

This paper proves the existence of a limiting distribution for free path lengths on flat surfaces with small circular obstacles and relates it to the heights in zippered rectangle decompositions.

Contribution

It establishes the limiting distribution of free path lengths as obstacle radius approaches zero and connects it to zippered rectangle decompositions.

Findings

Limiting distribution exists as obstacle radius tends to zero

Distribution relates to heights in zippered rectangle decompositions

Provides a new link between geometric and dynamical properties

Abstract

In this note, we prove the existence of a limiting distribution of the free path lengths on flat surfaces with circular obstacles as the radius of the obstacles goes to zero. Moreover, we relate this distribution to the distribution of the heights of zippered rectangle decompositions of flat surfaces.