{"paper":{"title":"On a recursive construction of Dirichlet form on the Sierpi\\'nski gasket","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Hua Qiu, Ka-Sing Lau, Qingsong Gu","submitted_at":"2017-07-05T15:04:47Z","abstract_excerpt":"Let $\\Gamma_n$ denote the $n$-th level Sierpi\\'nski graph of the Sierpi\\'nski gasket $K$. We consider, for any given conductance $(a_0, b_0, c_0)$ on $\\Gamma_0$, the Dirchlet form ${\\mathcal E}$ on $K$ obtained from a recursive construction of compatible sequence of conductances $(a_n, b_n, c_n)$ on $\\Gamma_n, n\\geq 0$. We prove that there is a dichotomy situation: either $a_0= b_0 =c_0$ and ${\\mathcal E}$ is the standard Dirichlet form, or $a_0 >b_0 =c_0$ (or the two symmetric alternatives), and ${\\mathcal E}$ is a non-self-similar Dirichlet form independent of $a_0, b_0$. The second situatio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.01426","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"}