Complexity for billiards in regular N-gons
Jayadev Athreya, Pascal Hubert, and Serge Troubetzkoy
TL;DR
This paper calculates the complexity of billiard trajectories in regular N-gons and similar polygons, providing new counting results for saddle connections on lattice surfaces to answer an open question.
Contribution
It introduces a novel counting method for saddle connections on lattice surfaces, advancing understanding of billiard dynamics in rational polygons.
Findings
Computed the billiard language complexity for regular N-gons.
Provided a new counting result for saddle connections by combinatorial length.
Answered an open question by Cassaigne-Hubert-Troubetzkoy.
Abstract
We compute the complexity of the billiard language of the regular Euclidean -gons (and other families of rational lattice polygons), answering a question posed by Cassaigne-Hubert-Troubetzkoy. Our key technical result is a counting result for saddle connections on lattice surfaces, when we count by combinatorial length.
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