{"paper":{"title":"Order symmetry and orthogonality of trajectories in discrete interval exchange transformations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.CO","authors_text":"Luca Q. Zamboni, S\\'ebastien S. Ferenczi","submitted_at":"2026-07-04T09:17:45Z","abstract_excerpt":"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 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.03785","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.03785/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}