{"paper":{"title":"Phase transition on Exel crossed products assocaited to dilation matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Iain Raeburn, Jacqui Ramagge, Marcelo Laca","submitted_at":"2011-01-25T03:52:11Z","abstract_excerpt":"An integer matrix $A\\in M_d(\\Z)$ induces a covering $\\sigma_A$ of $\\T^d$ and an endomorphism $\\alpha_A:f\\mapsto f\\circ \\sigma_A$ of $C(\\T^d)$ for which there is a natural transfer operator $L$. In this paper, we compute the KMS states on the Exel crossed product $C(\\T^d)\\rtimes_{\\alpha_A,L}\\N$ and its Toeplitz extension. We find that $C(\\T^d)\\rtimes_{\\alpha_A,L}\\N$ has a unique KMS state, which has inverse temperature $\\beta=\\log|\\det A|$. Its Toeplitz extension, on the other hand, exhibits a phase transition at $\\beta=\\log|\\det A|$, and for larger $\\beta$ the simplex of KMS$_\\beta$ states is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.4713","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"}