Order symmetry and orthogonality of trajectories in discrete interval exchange transformations

Abstract

Let $\pi=(<_D,<_A)$ 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 $\pi$ are characterized by the following order symmetry : $v_i^\omega <_D v_j^\omega$ (lexicographically) if and only if $v_i^{-\omega}<_Av_j^{-\omega}$ (reverse lexicographically). For general words $u$ and $v$ over $A$, the orders need not agree in which case either $u^\omega<_A v^\omega<_D u^\omega$ (Type 1) or $v^\omega<_A u^\omega<_D v^\omega$ (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 $\lambda(u)$ and $\lambda(v).$ We show that $|T_2(u,v)|-|T_1(u,v)|=\lambda(u)^T \Omega \lambda(v)$ where $\Omega$ is a $k\times k$ skew symmetric matrix depending only on $\pi.$ It follows that the Parikh vectors of the trajectories of a discrete interval exchange are orthogonal with respect to $\Omega.$ 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 $d=\dim \ker (\Omega).$ 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.