Boundary distance, lens maps and entropy of geodesic flows of Finsler metrics

TL;DR

This paper demonstrates that small boundary and lens map perturbations of simple Finsler metrics can be realized by actual Finsler metrics, and uses this to create a sphere with a geodesic flow exhibiting positive metric entropy and local entropy generation.

Contribution

It proves that small boundary and lens map perturbations correspond to genuine Finsler metrics and constructs a sphere with a geodesic flow of positive metric entropy.

Findings

Small boundary distance perturbations correspond to actual Finsler metrics.

Small lens map perturbations are realizable by Finsler metrics.

Constructed a sphere with a geodesic flow exhibiting positive metric entropy.

Abstract

We show that a small perturbation of the boundary distance function of a simple Finsler metric on the $n$-disc is also the boundary distance function of some Finsler metric. (Simple metric form an open class containing all flat metrics.) The lens map is map that sends the exit vector to the entry vector as a geodesic crosses the disc. We show that a small perturbation of a lens map of a simple Finsler metric is in its turn the lens map of some Finsler metric. We use this result to construct a smooth perturbation of the metric on the standard 4-dimensional sphere to produce positive metric entropy of the geodesic flow. Furthermore, this flow exhibits local generation of metric entropy, that is positive entropy is generated in arbitrarily small tubes around one trajectory.