Hypercubic Billiard Words

• Hypercubic billiard words are infinite symbolic sequences encoding the hyperfaces encountered by a billiard trajectory in a d-dimensional unit hypercube, generalizing Sturmian words.
• They are constructed using dynamical systems techniques such as billiard maps, toral translations, and cut-and-project methods, yielding a complexity growth asymptotically proportional to n^(3d-3).
• Their balance properties and collapsing binomial complexities highlight deep connections with aperiodic order, quasicrystal mathematics, and higher-dimensional Christoffel analogues.

Hypercubic billiard words are infinite symbolic sequences that encode the sequence of hyperfaces or faces encountered by a billiard trajectory in a high-dimensional unit hypercube. These words serve as generalizations of Sturmian words for higher dimensional settings, arising naturally in geometric, dynamical, combinatorial, and aperiodic order contexts through codings of billiard flows, toral translations, and cut-and-project constructions.

1. Dynamical and Symbolic Construction

The billiard map in a hypercube $C = [0,1]^d \subset \mathbb{R}^d$ is defined on the phase space

$\partial C \times \mathbb{RP}^{d-1},$

where $\partial C$ denotes the boundary of the hypercube and $\mathbb{RP}^{d-1}$ is the real projective space of directions. A billiard ball follows a straight-line until it hits a face and reflects according to the specular reflection law. The symbolic coding associates to each (unoriented) face family a symbol from a $d$-letter alphabet $\mathcal{A}$. The infinite word records, in order, the faces intersected by the orbit, typically assigning the same symbol to each pair of parallel hyperfaces. The “unfolding” method—reflected images of the cube in $\mathbb{R}^d$—translates the billiard flow to a straight-line flow on the periodic tiling, thereby mapping trajectories to symbolic codings via their successive intersections with the hypercube faces (Bedaride et al., 2011).

Alternatively, in the cut-and-project framework, the straight-line flow is projected onto a lower-dimensional torus by selecting an irrational direction $\rho = (1, \theta_1, \ldots, \theta_d)$, yielding a minimal (uniquely ergodic) translation. The coding of the trajectory as it traverses an “exchange of pieces” partition (the projection of $C$ into a “window” $W$) recovers the hypercubic billiard word (Bédaride et al., 25 Feb 2025).

2. Complexity Functions and Combinatorics

The core combinatorial invariant of a hypercubic billiard language is its complexity function $p(n, d)$, defined as the number of distinct length-$n$ factors (words) obtainable by coding billiard trajectories. The principal result in the case of the $d$-dimensional hypercube is that

$$\exists C_1,C_2>0 : \quad C_1 n^{3d-3} \leq p(n,d) \leq C_2 n^{3d-3},$$

that is, the complexity grows asymptotically as $n^{3d-3}$ for large $n$ (Bedaride et al., 2011). This exponent arises from a detailed enumeration of generalized diagonals (defined as trajectory segments between lower-dimensional faces), which generate new words in the symbolic language as their combinatorial type varies.

In the two-dimensional setting (the square), the complexity reduces to the Sturmian formula $p(n) = n+1$; for the three-dimensional cube, the complexity for B-irrational directions is $p(n, \omega) = n^2 + n + 1$ for $n > 0$ (Bedaride, 2012).

The complexity may also be interpreted via partition refinements,

$$\mathcal{P}_n = \bigvee_{i=0}^n T^{-i} \mathcal{P},$$

where $\mathcal{P}$ is the initial partition of the face, and $T$ is the billiard map. The difference sequence $s(n) = p(n+1) - p(n)$ is controlled combinatorially by sums over bispecial words:

$s(n+1, d) - s(n, d) = \sum_{v \in BL(n, d)} i(v),$

where the index $i(v)$ encapsulates bi-, left- and right-extensions (Bedaride et al., 2011).

3. Balance Properties and Factor Discrepancy

Hypercubic billiard words are distinguished by their balance behavior:

• Letter-level balancedness: For each alphabet symbol $a$, the difference in the number of occurrences of $a$ between any two length-$n$ factors is uniformly bounded, independently of $n$. This outcome follows from the unique ergodicity of the associated toral translation and the bounded remainder set (BRS) property for the cylinder sets corresponding to single symbols (Bédaride et al., 25 Feb 2025).
• Factor-level unbalancedness: For $d \geq 2$, there exist factors (words of length at least 2) for which the discrepancy

$D(n, w) = S_n(1_w) - n\mu[w]$

(where $S_n(1_w)$ counts occurrences of $w$ in the first $n$ letters, and $\mu[w]$ is its frequency) grows unbounded as $n$ increases. This reflects the failure of the BRS property for two-letter blocks and longer factors.

The proof leverages cohomological methods (specifically, failure of certain cocycles to be coboundaries in the dynamical system) and arithmetic characterizations of BRS via geometric properties of the window sets arising in the cut-and-project representation. In particular, for the cubic (3-letter) case, no factor of length 2 or more can be balanced (Bédaride et al., 25 Feb 2025).

4. Binomial Complexities and Collapse Phenomena

Complexity of infinite words can be refined using $k$-binomial complexity $b_w^k(n)$, counting the number of equivalence classes induced by equality of scattered subword multiplicities up to length $k$:

$$u \sim_k v \iff \forall x: |x| \leq k,~ \binom{u}{x} = \binom{v}{x}$$

with $\binom{u}{x}$ the number of scattered occurrences of $x$ in $u$. For hypercubic billiard words, the $k$-binomial complexity is known to “collapse” at $k=2$:

$b_w^{2}(n) = p_w(n) \quad \forall n,$

i.e., for $k \geq 2$ the $k$-binomial complexity coincides with the subword complexity (Vivion, 14 Sep 2025). This collapse property is shared by all $d$-ary 1-balanced words, words of minimal subword complexity, and certain colored Sturmian constructions, and is explained by the 1-balanced nature of all binary projections of the hypercubic billiard word.

5. Geometric and Algebraic Analogues

Hypercubic billiard words are closely tied to higher-dimensional analogues of Christoffel words. In “A d-dimensional extension of Christoffel words” (Labbé et al., 2014), Christoffel graphs on $\mathbb{Z}^d$ generalize the notion to higher-dimensional periodic directed subgraphs encoding the monotonicity of linear forms. These graphs exhibit symmetry, conjugation under flips and translations (a higher-dimensional Pirillo property), and natural embeddings into $(d-1)$-tori, mirroring the periodicity and symmetry seen in billiard codings. The Christoffel graph is fully characterized by the property that it is conjugate (by translation) to its flip—this generalizes central palindromic properties from Sturmian words to the hypercubic context.

Connections to affine Weyl groups are established via random billiard walks, where the geometric position after $K$ random subword-induced transitions (with independent deletion probability $1-p$) converges—after normalization—to a central limit, with covariance reflecting the underlying Coxeter geometry and, in type $\widetilde{A}_r$, parallel to hypercubic structure (Defant et al., 19 Jan 2025).

6. Discrete Integrable Models and Lattice Dynamics

Discrete integrable systems arising from billiards in confocal quadrics extend the framework beyond cubic lattices, leading to more intricate honeycomb tessellations by rectified hypercubes and cross polytopes (Radnovic, 2015). In these models, the combinatorial encoding of trajectories—the hypercubic billiard words—can be constructed via assignments of hyperplanes (or their intersections) to lattice sites according to discrete billiard reflection rules. The underlying structures govern the word generation, capturing both the geometric and combinatorial facets of the dynamics.

7. Aperiodic Order, Quasicrystals, and Applications

Hypercubic billiard words serve as paradigms for one-dimensional quasicrystals, being cut-and-project sets with cubical windows. They exhibit hierarchical structural properties reminiscent of those seen in quasicrystal tilings, and their symbolic and combinatorial characterizations provide insight into low-complexity, aperiodically ordered dynamical systems. The paper of balance and unbalanced properties, cohomological invariants, and complexity collapses has implications for ergodic theory, tiling theory (notably, links with Nivat’s conjecture), combinatorics on words, and pattern matching algorithms (Bédaride et al., 25 Feb 2025, Vivion, 14 Sep 2025).

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In summary, hypercubic billiard words encapsulate the intricate interplay between geometric dynamical systems, combinatorial word theory, and aperiodic order. Their symbolic codings, complexity growth rates, balance properties, and algebraic-geometric analogues position them as central objects in the broader paper of multidimensional symbolic dynamics and quasicrystal mathematics.