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The sets are genuinely non-hyperbolic containing intermingled horseshoes of different hyperbolic behavior (contracting and expanding center).\n  We prove that for every $k\\ge 1$ there is a diffeomorphism $F$ with a transitive set $\\Lambda$ as above such that the pressure map $P(t)=P(t\\, \\varphi)$ of the potential $\\varphi= -\\log \\,\\l"},"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":"1303.0581","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-03-03T23:35:27Z","cross_cats_sorted":[],"title_canon_sha256":"e8fbfd0e0a994cb2af14d29a6239932596ad5bf3807ab62b41c1d8eb49801b31","abstract_canon_sha256":"61ebfceb6eb073dad8441077b66c76e67b81130b65c81811743ebb87a5cf861e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:51:17.620301Z","signature_b64":"eaLPqM4I+Yh/JKjaxweC172ipVkEVz67OrUWyRLasci7THHvAQHhyVWUFFHyg1V49HiVplvS/YXAyGMoYECXAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1588b1dcbd170fd35e031f6313268bb353f2342241b9b318cb02dc6379475f09","last_reissued_at":"2026-05-18T01:51:17.619718Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:51:17.619718Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Abundant rich phase transitions in step skew products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"K. 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