{"paper":{"title":"Justifications of spatial entropies of multi-dimensional symbolic dynamical systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Song-Sun Lin, Wen-Guei Hu","submitted_at":"2014-12-22T01:32:14Z","abstract_excerpt":"The commonly used spatial entropy $h_{r}(\\mathcal{U})$ of the multi-dimensional shift space $\\mathcal{U}$ is the limit of growth rate of admissible local patterns on finite rectangular sublattices which expands to whole space $\\mathbb{Z}^{d}$, $d\\geq 2$. This work studies spatial entropy $h_{\\Omega}(\\mathcal{U})$ of shift space $\\mathcal{U}$ on general expanding system $\\Omega=\\{\\Omega(n)\\}_{n=1}^{\\infty}$ where $\\Omega(n)$ is increasing finite sublattices and expands to $\\mathbb{Z}^{d}$. $\\Omega$ is called genuinely $d$-dimensional if $\\Omega(n)$ contains no lower-dimensional part whose size "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.6859","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":""},"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"}