{"paper":{"title":"Automatic actions I. Bounded automata and orbits","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Laurent Bartholdi","submitted_at":"2026-06-06T11:06:57Z","abstract_excerpt":"We develop the theory of \"automatic actions\": (semi)groups acting by $\\omega$-regular transformations on an $\\omega$-regular language, showing that it covers a large class of heretofore-unrelated examples.\n  We focus on the subclass of actions by \"bounded\" $\\omega$-regular transformations, those for which the B\\\"uchi automata encoding the action do not have two connected non-trivial cycles.\n  We show that, for bounded actions of inverse semigroups, the orbit relation is also $\\omega$-regular.\n  We deduce a number of corollaries, in particular decidability, for such actions, of minimality, topo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08101","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/2606.08101/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"}