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Hausdorff and Packing Measure of Sets of Generic Points: A Zero‐Infinity Law
Author(s) -
Ma Jihua,
Wen Zhiying
Publication year - 2004
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/s0024610703005040
Subject(s) - mathematics , hausdorff measure , gibbs measure , measure (data warehouse) , outer measure , zero (linguistics) , infinity , hausdorff distance , probability measure , σ finite measure , hausdorff space , metric space , combinatorics , metric (unit) , hausdorff dimension , discrete mathematics , minkowski–bouligand dimension , mathematical analysis , fractal , fractal dimension , linguistics , philosophy , computer science , operations management , database , economics
On the symbolic space endowed with a metric given by a Gibbs measure, it is shown that, for any invariant probability measure μ other than the given Gibbs measure, the set of μ‐generical points satisfies a ‘zero‐infinity law’ (in particular, its Hausdorff and packing measure are infinite). This extends a result of R. Kaufman on Besicovitch–Eggleston sets, and applies to level sets of Birkhoff averages and certain subsets of self‐similar sets.