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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1707.01426","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-07-05T15:04:47Z","cross_cats_sorted":[],"title_canon_sha256":"19581fc339895f6b0d05bd2ea53f6e916989dd9a4a6752a53f80b8e79766f19a","abstract_canon_sha256":"0e418e880e4f0065fea02396eefdd031190cfee68af29fa3e3e916296001a87f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:52.354240Z","signature_b64":"5cuL3+0CJAC30f6aMC4Ivzg2uSa407+/tG/PsBUltsD/lFsMLTvrpvUd0C1lde9uVDcMmZbEX4iSJbnpMUoVCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"16dc524621250db715aff8aaf937d77cc8d6c1e6125ecdda339dad2a40b6772d","last_reissued_at":"2026-05-18T00:40:52.353684Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:52.353684Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $a_0, b_0$. 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