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Hausdorff and Conformal Measures on Julia Sets with a Rationally Indifferent Periodic Point
Author(s) -
Denker M.,
Urbański M.
Publication year - 1991
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-43.1.107
Subject(s) - hausdorff dimension , mathematics , julia set , hausdorff measure , conformal map , measure (data warehouse) , uniqueness , outer measure , zero (linguistics) , exponent , pure mathematics , dimension function , function (biology) , dimension (graph theory) , packing dimension , combinatorics , mathematical analysis , fractal , minkowski–bouligand dimension , fractal dimension , linguistics , philosophy , database , evolutionary biology , biology , computer science
We show that the Hausdorff dimension δ of a non‐hyperbolic Julia set J (T) without critical points can be expressed by the smallest zero of the pressure function t ↦ P (T, − t log |T′|). This result is similar to the Bowen‐Manning‐McCluskey formula. The Hausdorff dimension is also shown to be the smallest exponent t εR for which a t ‐conformal measure in the sense of Sullivan exists. We also prove uniqueness properties of t ‐conformal measures, and we prove the absolute continuity of the Hausdorff measure H δ with respect to any δ‐conformal measure.