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We prove that $f$ is topologically mixing and if $c>1/4$ then $f$ is mixing with respect to Lebesgue measure. Furthermore we prove that the speed of mixing is exponential."},"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":"1202.2162","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-02-10T00:41:45Z","cross_cats_sorted":[],"title_canon_sha256":"637848f78e6f1b949548f87509344fc3b67c4fc8628b291f4f33a638c83edd2b","abstract_canon_sha256":"62e7bba0777988c5d8e0cfeb1a04fbc2837ded2b24fb9ab331ee43c1d79c9dfd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:58:31.457814Z","signature_b64":"x+J9PtDwuACn20CqEOn6+dolsUXcGL7lk5jjxKoRC5wTseuIsoVhMZmFa4J3Bfo2AHKk48CBC4ud1cmo85HbBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"58d61222676d999866256dd48312230a9a8f9967ec98b44da6e441b028b45a53","last_reissued_at":"2026-05-18T01:58:31.457262Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:58:31.457262Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Exponential speed of mixing for skew-products with singularities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"J. Vieitez, M. J. Pacifico, R. Markarian","submitted_at":"2012-02-10T00:41:45Z","abstract_excerpt":"Let $f: [0,1]\\times [0,1] \\setminus {1/2} \\to [0,1]\\times [0,1]$ be the $C^\\infty$ endomorphism given by $$f(x,y)=(2x- [2x], y+ c/|x-1/2|- [y+ c/|x-1/2|]),$$ where $c$ is a positive real number. We prove that $f$ is topologically mixing and if $c>1/4$ then $f$ is mixing with respect to Lebesgue measure. 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