Hypercubic billiard word

Formal definition

Formally, let

• w be a scale word with signature a₁X₁, ..., a_rX_r (i.e. w is a scale word with a_i-many X_i steps);
• n = a₁ + ... + a_r be the length of w;
• L be a line of the form L(t) = (a₁, ..., a_r)t + v₀, where v₀ is a constant vector in R^r. We say that L is in generic position if L intersects the hyperplane x₁ = 0 at a point (0, α₁, α₂, ... α_r-1) where α_i and α_j/α_i for i ≠ j are irrational.

We say that w is a rank-r billiard scale if any line in generic position of the form (a₁, ..., a_r)t + v₀ has intersections with coordinate level planes x_i = k ∈ Z that spell out the scale as you move in the positive t direction along that line. The term billiard scale is motivated by considering a point particle (a "billiard ball") that moves in an r-dimensional cubic room [0, 1]^r and bounces off the room's (r − 1)-dimensional walls. If the particle initial velocity's is given by writing the scale signature as the vector (a₁, ..., a_r), the billiard's trajectory will be periodic. If the particle's initial position is outside a certain measure 0 set, any collision will only be with one wall at a time. The pattern of which walls the particle collides with then spells out a billiard scale.

Properties

Proofs to be added

• Mosses are rank-2 billiard scales
• A billiard scale projects to a billiard scale of lower rank when one removes all instances of some subset of its step sizes