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 !-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.