On non-periodic and non-dense billiard trajectories Part 1 and Part 2

TL;DR

This paper proves that Galperin's widely accepted method for constructing non-dense, non-periodic billiard trajectories in triangles is fundamentally flawed, as all such constructions are actually periodic, invalidating previous assumptions.

Contribution

It demonstrates that Galperin's method cannot produce non-periodic trajectories, challenging a long-standing approach in billiard dynamics.

Findings

Galperin's method always yields periodic trajectories

All potential examples from Galperin's method are periodic

The method cannot produce non-dense, non-periodic trajectories

Abstract

This paper shows that the method of Galperin which had been widely accepted in the literature since 1983 for constructing a non-dense and non-periodic trajectory in a triangle can never work. This is done by showing that all possible examples that can be produced by Galperin's method of which there are infinitely many all produce a periodic trajectory.