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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  and
ZHIYING WEN
JIHUA MA
Affiliation:
Department of Mathematics, Wuhan University, Wuhan 430072, China Department of Mathematics, Friedrich-Schiller University, Jena D-07743, [email protected]
ZHIYING WEN
Affiliation:
Department of Mathematics, Tsinghua University, Beijing 10084, [email protected]
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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.

Keywords

28A78 (primary)58F03 (secondary)
Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 69 , Issue 2 , April 2004 , pp. 383 - 406
Copyright
The London Mathematical Society 2004

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