Anosov geodesic flows, billiards and linkages

TL;DR

This paper demonstrates that under certain conditions, the geodesic flow on a flattened surface approximates billiard flow and can be Anosov, with applications to mechanical linkages exhibiting chaotic behavior.

Contribution

It establishes a link between flattened surfaces and billiard dynamics, providing explicit examples of linkages with Anosov behavior.

Findings

Geodesic flow converges to billiard flow under flattening.

Dispersive billiards with finite horizon lead to Anosov geodesic flows.

Explicit edge lengths for a new Anosov linkage example.

Abstract

Any smooth surface in R^3 may be flattened along the z-axis, and the flattened surface becomes close to a billiard table in R^2 . We show that, under some hypotheses, the geodesic flow of this surface converges locally uniformly to the billiard flow. Moreover, if the billiard is dispersive and has finite horizon, then the geodesic flow of the corresponding surface is Anosov. We apply this result to the theory of mechanical linkages and their dynamics: we provide a new example of a simple linkage whose physical behavior is Anosov. For the first time, the edge lengths of the mechanism are given explicitly.