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HAUSDORFF AND PACKING MEASURE OF SETS OF GENERIC POINTS: A ZERO-INFINITY LAW

Published online by Cambridge University Press:  29 March 2004

JIHUA MA
Affiliation:
Department of Mathematics, Wuhan University, Wuhan 430072, China Department of Mathematics, Friedrich-Schiller University, Jena D-07743, Germanylaoma2001@yahoo.com
ZHIYING WEN
Affiliation:
Department of Mathematics, Tsinghua University, Beijing 10084, Chinawenzy@tsinghua.edu.cn
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Abstract

On the symbolic space endowed with a metric given by a Gibbs measure, it is shown that, for any invariant probability measure $\mu$ other than the given Gibbs measure, the set of $\mu$-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.

Type
Notes and Papers
Copyright
The London Mathematical Society 2004

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