Order symmetry and orthogonality of trajectories in discrete interval exchange transformations

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 $v_i^{-ω}<_Av_j^{-ω}$ (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 $|T_2(u,v)|-|T_1(u,v)|=λ(u)^T Ωλ(v)$ 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 $\lfloor \frac{k+d}2 \rfloor$ on the number of distinct trajectories where $d=\dim \ker (Ω).$ If $T$ is symmetric, then the number of distinct trajectories is at most$\lfloor \frac{k+1}2 \rfloor.$ An alternate interpretation of this result is that on an ordered $k$-letter alphabet, there are at most $\lfloor \frac{k+1}2 \rfloor$ 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.