Uniqueness of billiard coding in polygons

Abstract

We consider polygonal billiards and we show that every nonperiodic billiard trajectory hits a unique sequence of sides if all the holes of the polygonal table have non-zero minimal diameters, generalizing a classical theorem of Galperin, Krüger and Troubetzkoy. Our approach uses symbolic dynamics and elementary geometry. We review some classical constructions in polygonal billiards and we introduce, as one of our main tools, a method to code pairs of parallel billiard trajectories in non-simply connected polygons. We also discuss some useful properties of ‘generalized trajectories’, which can be uniquely constructed from the limits of converging sequences of billiard codings.