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We prove also that if $\\limsup |t_{m+1}|^{1/q_m}<1$ and $q_m\\to \\infty$, then the Julia set of $f_c$ is not locally connected and the Mandelbrot set is locally connected at $c$ provided that all the renormalizations are non-primitive (satellite). This quantifies a construction of A. Douady and J. Hubbard, and weakens a condition proposed"},"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":"0710.4406","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2007-10-24T08:41:47Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"3e7fb5a2e58bc4cdf5115443537311ea9d877856e0046a257f5c7ab59f3f16f1","abstract_canon_sha256":"19c91eabcc854cfd5e3fb71b53209be50d7519058b34f1e547fdb07513cc4b29"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:24:57.794687Z","signature_b64":"N5IdsoT76cr4FS+285ezC5KXD8HW65DqKStwWQY1jOhxjDG3XxV4Wvu1rE2tg6O8JTwBrhiY8rHkbc/I0c/jCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2f9230eb0dbb7c0e95c2b3420c704aded5a579bd189ab0052250b77e8eb69554","last_reissued_at":"2026-05-18T02:24:57.793990Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:24:57.793990Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rigidity and non local connectivity of Julia sets of some quadratic polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.DS","authors_text":"Genadi Levin","submitted_at":"2007-10-24T08:41:47Z","abstract_excerpt":"For an infinitely renormalizable quadratic map $f_c: z\\mapsto z^2+c$ with the sequence of renormalization periods ${k_m}$ and rotation numbers ${t_m=p_m/q_m}, we prove that if $\\limsup k_m^{-1}\\log |p_m|>0$, then the Mandelbrot set is locally connected at $c$. We prove also that if $\\limsup |t_{m+1}|^{1/q_m}<1$ and $q_m\\to \\infty$, then the Julia set of $f_c$ is not locally connected and the Mandelbrot set is locally connected at $c$ provided that all the renormalizations are non-primitive (satellite). This quantifies a construction of A. Douady and J. 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