{"paper":{"title":"Ratio geometry, rigidity and the scenery process for hyperbolic Cantor sets","license":"","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Albert M. Fisher, Tim Bedford","submitted_at":"1994-05-31T00:00:00Z","abstract_excerpt":"Given a $C^{1+\\gamma}$ hyperbolic Cantor set $C$, we study the sequence $C_{n,x}$ of Cantor subsets which nest down toward a point $x$ in $C$. We show that $C_{n,x}$ is asymptotically equal to an ergodic Cantor set valued process. The values of this process, called limit sets, are indexed by a H\\\"older continuous set-valued function defined on D. Sullivan's dual Cantor set. We show the limit sets are themselves $C^{k+\\gamma}, C^\\infty$ or $C^\\omega$ hyperbolic Cantor sets, with the highest degree of smoothness which occurs in the $C^{1+\\gamma}$ conjugacy class of $C$. The proof of this leads t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9405217","kind":"arxiv","version":1},"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"}