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Suppose the products of the jump ratios of $Df_1$ and $Df_2$ do not coincide, i.e. $\\frac{Df_1(a_1-0)}{Df_1(a_1+0)}\\times \\frac{Df_1(b_1-0)}{Df_1(b_1+0)}\\neq \\frac{Df_2(a_2-0)}{Df_2(a_2+0)}\\times \\frac{Df_2(b_2-0)}{Df_2(b_2+0)}$. Then the map $\\psi$ conjugating $f_1$ and $f_2$ is a singul"},"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":"1110.6125","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-10-27T15:51:37Z","cross_cats_sorted":[],"title_canon_sha256":"9864b57a04bf90b29ecf253f4bd6d92a3a7366063a498a1d69e6c51dbb583480","abstract_canon_sha256":"5a6d3ad693a065cdb272871233493a13923d25ad4cb8f423a581a0ac185ea0c4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:53:19.150073Z","signature_b64":"tcs5EZVhvRR8Y/9+g3kp/ZIcyA1CfQ4tROYeqE4CP6WeLbKEhDlvH1vZMpu4mbzn4y6VLIhwEkHVrHO1K7QBCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eee819e3f07c42b7be2921fe772a0c8aad12a372f6cb34bb635f7bda7e7c3cf5","last_reissued_at":"2026-05-17T23:53:19.149422Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:53:19.149422Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On conjugations of circle homeomorphisms with two break points","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Akhtam Dzhalilov, Dieter Mayer, Habibulla Akhadkulov","submitted_at":"2011-10-27T15:51:37Z","abstract_excerpt":"Let $f_i\\in C^{2+\\alpha}(S^1\\setminus \\{a_i,b_i\\}), \\alpha >0, i=1,2$ be circle homeomorphisms with two break points $a_i,b_i$, i.e. discontinuities in the derivative $f_i$, with identical irrational rotation number $rho$ and $\\mu_1([a_1,b_1])= \\mu_2([a_2,b_2])$, where $\\mu_i$ are invariant measures of $f_i$. Suppose the products of the jump ratios of $Df_1$ and $Df_2$ do not coincide, i.e. $\\frac{Df_1(a_1-0)}{Df_1(a_1+0)}\\times \\frac{Df_1(b_1-0)}{Df_1(b_1+0)}\\neq \\frac{Df_2(a_2-0)}{Df_2(a_2+0)}\\times \\frac{Df_2(b_2-0)}{Df_2(b_2+0)}$. 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