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Journal of the London Mathematical Society (1)

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HAUSDORFF AND PACKING MEASURE OF SETS OF GENERIC POINTS: A ZERO-INFINITY LAW
JIHUA MA, ZHIYING WEN
Journal:

Journal of the London Mathematical Society / Volume 69 / Issue 2 / April 2004

Published online by Cambridge University Press:

29 March 2004, pp. 383-406

Print publication:

April 2004
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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.