Order symmetry and orthogonality of trajectories in discrete interval exchange transformations

Sébastien S. Ferenczi; Luca Q. Zamboni

paper

Order symmetry and orthogonality of trajectories in discrete interval exchange transformations

arXiv:2607.03785

Abstract

Let $π = (<_{D}, <_{A})$ be a pair of distinct orders on a $k$ -letter alphabet $A .$ The periodic trajectories $v_{i}^{\infty}$ of a discrete $k$ -interval exchange transformations $T$ with permutation $π$ are characterized by the following order symmetry : $v_{i}^{ω} <_{D} v_{j}^{ω}$ (lexicographically) if and only if $v_{i}^{- ω} <_{A} v_{j}^{- ω}$ (reverse lexicographically). For general words $u$ and $v$ over $A$ , the orders need not agree in which case either $u^{ω} <_{A} v^{ω} <_{D} u^{ω}$ (Type 1) or $v^{ω} <_{A} u^{ω} <_{D} v^{ω}$ (Type 2). We partition all such order crossings amongst the set of conjugates of two words $u$ and $v$ into disjoint families $T_{1} (u, v)$ and $T_{2} (u, v)$ and define the index $i (u, v)$ by $∣ T_{1} (u, v) ∣ + ∣ T_{2} (u, v) ∣.$ Remarkably the difference, $∣ T_{2} (u, v) ∣ - ∣ T_{1} (u, v) ∣,$ depends only on the Parikh vectors $λ (u)$ and $λ (v) .$ We show that $∣ T_{2} (u, v) ∣ - ∣ T_{1} (u, v) ∣ = λ (u)^{T} Ω λ (v)$ where $Ω$ is a $k \times k$ skew symmetric matrix depending only on $π .$ It follows that the Parikh vectors of the trajectories of a discrete interval exchange are orthogonal with respect to $Ω.$ Applied to dimension $3,$ we obtain an arithmetic formula for the number of orbits in a discrete $3$ -interval exchange and hence a characterization of minimality. For general $k,$ the orthogonality of the trajectories gives an upper bound $⌊ \frac{k + d}{2} ⌋$ on the number of distinct trajectories where $d = dim ker (Ω) .$ If $T$ is symmetric, then the number of distinct trajectories is at most $⌊ \frac{k + 1}{2} ⌋ .$ An alternate interpretation of this result is that on an ordered $k$ -letter alphabet, there are at most $⌊ \frac{k + 1}{2} ⌋$ primitive, pairwise non conjugate perfectly clustering words which perfectly cluster collectively in a single array in which all their conjugates are arranged in increasing order.