{"paper":{"title":"Nonexpansive Z^2 subdynamics and Nivat's conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Bryna Kra, Van Cyr","submitted_at":"2012-08-20T19:45:37Z","abstract_excerpt":"For a finite alphabet $\\A$ and $\\eta\\colon \\Z\\to\\A$, the Morse-Hedlund Theorem states that $\\eta$ is periodic if and only if there exists $n\\in\\N$ such that the block complexity function $P_\\eta(n)$ satisfies $P_\\eta(n)\\leq n$, and this statement is naturally studied by analyzing the dynamics of a $\\Z$-action associated to $\\eta$. In dimension two, we analyze the subdynamics of a $\\ZZ$-action associated to $\\eta\\colon\\ZZ\\to\\A$ and show that if there exist $n,k\\in\\N$ such that the $n\\times k$ rectangular complexity $P_{\\eta}(n,k)$ satisfies $P_{\\eta}(n,k)\\leq nk$, then the periodicity of $\\eta$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.4090","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}