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These sets are shown to provide fractal models of radial limit sets of normal subgroups of Kleinian groups of Schottky type. Our main result states that if $\\Phi$ is symmetric and linear, then we have that $\\dim_{H}(\\Lr(N,\\Phi))=\\dim_{H} \\Lr(\\F_d,\\Phi))$ if and only if the quotient group $\\F_d /N$ is amenable, where $\\dim_{H}$ denotes the Hausdorff dimension. This extends a result of Br"},"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":"1106.0026","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-05-31T20:59:06Z","cross_cats_sorted":[],"title_canon_sha256":"43c6a54e0f4e612e19e1b358397fab8cd02527051829ea41290b1ad02327c697","abstract_canon_sha256":"224a27d88e1cadca4c4926a93693ba40986d61b0b328ea046622af7d85b6f284"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:27:16.204879Z","signature_b64":"RTN0hyn09GI42cyvn5e9M6nkH5R7Qfr4I0Iv3whR3DrvPwQ7ZlvMEyvSydNh00UVyHTzGf9WpUZQGywqLxkxBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e02311c1e3fa2cd023ea1a3e46042976ee54bc4b35cc43a8a72223a81775e69a","last_reissued_at":"2026-05-18T01:27:16.204368Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:27:16.204368Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fractal Models for Normal Subgroups of Schottky Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Johannes Jaerisch","submitted_at":"2011-05-31T20:59:06Z","abstract_excerpt":"For a normal subgroup $N$ of the free group $\\F_d$ with at least two generators we introduce the radial limit set $\\Lr(N,\\Phi)$ of $N$ with respect to a graph directed Markov system $\\Phi$ associated to $\\F_d$. 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