Weak-mixing polygonal billiards

TL;DR

This paper demonstrates that within a dense set of polygons with fixed side orientations, the billiard flow exhibits weak-mixing behavior in almost every direction, revealing complex dynamical properties of such systems.

Contribution

It establishes the generic weak-mixing property for billiard flows in polygons with fixed combinatorics and only vertical or horizontal sides.

Findings

Weak-mixing occurs in a dense G δ subset of polygons.

Almost every direction exhibits weak-mixing behavior.

The result applies to polygons with fixed side orientations.

Abstract

We consider the set of polygons all of whose sides are vertical or horizontal with fixed combinatorics (for example all the figure "L"s). We show that there is a dense G $\delta$ subset of such polygons such that for each polygon in this G $\delta$ set the billiard flow is weakly-mixing in almost every direction.