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Suppose that $f'$ satisfies a certain Zygmund condition dependent on a parameter $\\gamma>0.$ We prove that the renormalizations of $f$ are approximated by M\\\"{o}bius transformations in $C^{1}$-norm if $\\gamma\\in (0,1]$ and they are approximated in $C^{2}$-norm if $\\gamma\\in (1,+\\infty).$ It is also shown, that the coefficients of M\\\"{o}bius transformations get asymptotically linearly dependent."},"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":"1510.03202","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-10-12T09:48:31Z","cross_cats_sorted":[],"title_canon_sha256":"f40447349037f5918ec2056ef97e399abc95f2f904b00acc3adf4e99da94780a","abstract_canon_sha256":"774557fee30d65ae1afcdb5875b24445ecfc90164f4a7c1d7755312b2bb36925"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:18:04.481547Z","signature_b64":"6UhOgGcpTNRzW4SJFAxRNW6p6NMcTt11xpepfSr25QFNR6FpBdNDgVKKNwqauUuOhzlgqqtGb148hIZ/RNpfDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a1fc6253ad4171eea8e65ff057e3916daa9fb5cfe4bc4526ea0120c247f881b2","last_reissued_at":"2026-05-18T01:18:04.480850Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:18:04.480850Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Renormalization of circle diffeomorphisms with a break-type singularity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Habibulla Akhadkulov, Mohd Salmi Md Noorani, Sokhobiddin Akhatkulov","submitted_at":"2015-10-12T09:48:31Z","abstract_excerpt":"Let $f$ be an orientation-preserving circle diffeomorphism with irrational rotation number and with a break point $\\xi_{0},$ that is, its derivative $f'$ has a jump discontinuity at this point. Suppose that $f'$ satisfies a certain Zygmund condition dependent on a parameter $\\gamma>0.$ We prove that the renormalizations of $f$ are approximated by M\\\"{o}bius transformations in $C^{1}$-norm if $\\gamma\\in (0,1]$ and they are approximated in $C^{2}$-norm if $\\gamma\\in (1,+\\infty).$ It is also shown, that the coefficients of M\\\"{o}bius transformations get asymptotically linearly dependent."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.03202","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1510.03202","created_at":"2026-05-18T01:18:04.480955+00:00"},{"alias_kind":"arxiv_version","alias_value":"1510.03202v2","created_at":"2026-05-18T01:18:04.480955+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.03202","created_at":"2026-05-18T01:18:04.480955+00:00"},{"alias_kind":"pith_short_12","alias_value":"UH6GEU5NIFY6","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_16","alias_value":"UH6GEU5NIFY65KHG","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_8","alias_value":"UH6GEU5N","created_at":"2026-05-18T12:29:44.643036+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/UH6GEU5NIFY65KHGL7YFPY4RNW","json":"https://pith.science/pith/UH6GEU5NIFY65KHGL7YFPY4RNW.json","graph_json":"https://pith.science/api/pith-number/UH6GEU5NIFY65KHGL7YFPY4RNW/graph.json","events_json":"https://pith.science/api/pith-number/UH6GEU5NIFY65KHGL7YFPY4RNW/events.json","paper":"https://pith.science/paper/UH6GEU5N"},"agent_actions":{"view_html":"https://pith.science/pith/UH6GEU5NIFY65KHGL7YFPY4RNW","download_json":"https://pith.science/pith/UH6GEU5NIFY65KHGL7YFPY4RNW.json","view_paper":"https://pith.science/paper/UH6GEU5N","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1510.03202&json=true","fetch_graph":"https://pith.science/api/pith-number/UH6GEU5NIFY65KHGL7YFPY4RNW/graph.json","fetch_events":"https://pith.science/api/pith-number/UH6GEU5NIFY65KHGL7YFPY4RNW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UH6GEU5NIFY65KHGL7YFPY4RNW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UH6GEU5NIFY65KHGL7YFPY4RNW/action/storage_attestation","attest_author":"https://pith.science/pith/UH6GEU5NIFY65KHGL7YFPY4RNW/action/author_attestation","sign_citation":"https://pith.science/pith/UH6GEU5NIFY65KHGL7YFPY4RNW/action/citation_signature","submit_replication":"https://pith.science/pith/UH6GEU5NIFY65KHGL7YFPY4RNW/action/replication_record"}},"created_at":"2026-05-18T01:18:04.480955+00:00","updated_at":"2026-05-18T01:18:04.480955+00:00"}