Parking garages with optimal dynamics

TL;DR

This paper introduces a new class of billiard tables called parking garages that exhibit optimal dynamical behavior similar to lattice surfaces, despite not being traditional polygons, highlighting novel challenges in their construction.

Contribution

It constructs generalized polygons with Veech dichotomy in billiard flow where the unfolded surface is not a lattice, revealing new phenomena in billiard dynamics.

Findings

Parking garages can satisfy Veech dichotomy without being lattice surfaces.

Construction of such parking garages involves overcoming significant geometric challenges.

The work highlights difficulties in creating genuine polygons with these properties.

Abstract

We construct generalized polygons (`parking garages') in which the billiard flow satisfies the Veech dichotomy, although the associated translation surface obtained from the Zemlyakov-Katok unfolding is not a lattice surface. We also explain the difficulties in constructing a genuine polygon with these properties.