On the Hausdorff measure of regular ω-languages in Cantor space
Ludwig Staiger
Abstract
This paper deals with the calculation of the Hausdorff measure of
regular ω-languages, that is, subsets of the Cantor
space definable by finite automata. Using methods for decomposing
regular ω-languages into disjoint unions of parts
of simple structure we derive two sufficient conditions under which
ω-languages with a closure definable by a finite
automaton have the same Hausdorff measure as this closure. The first
of these condition is related to the homogeneity of the local
behaviour of the Hausdorff dimension of the underlying set, and the
other with a certain topological density of the set in its closure.
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