Non-periodic not everywhere dense trajectories in triangular billiards

TL;DR

This paper constructs a family of isosceles triangles with non-periodic, non-dense trajectories, providing a definitive negative answer to a long-standing open question in triangular billiards.

Contribution

It introduces a new construction of triangles demonstrating non-dense, non-periodic trajectories, resolving a major open problem in the field.

Findings

Existence of non-dense, non-periodic trajectories in certain triangles

Construction method based on induced maps and interval exchange transformations

Negative answer to the density question in triangular billiards

Abstract

Building on tools that have been successfully used in the study of rational billiards, such as induced maps and interval exchange transformations, we provide a construction of a one-parameter family of isosceles triangles exhibiting non-periodic trajectories that are not everywhere dense. This provides, by elementary means, a definitive negative answer to a long-standing open question on the density of non-periodic trajectories in triangular billiards.