Hausdorff dimension of frequency sets of univoque sequences

摘要

We study the set Gamma consisting of univoque sequences, the set Gamma consisting of sequences in which the lengths of consecutive zeros and consecutive ones are bounded, and their frequency subsets Gamma(a), (Gamma) under bar (a), (Gamma) over bar (a) and Lambda(a) (Lambda) under bar (a), (Lambda) over bar (a), consisting of sequences respectively in Gamma and Lambda with frequency, lower frequency and upper frequency of zeros equal to some a is an element of [0, 1]. The Hausdorff dimension of all these sets are obtained by studying the dynamical system (Lambda((m)), sigma) where s is the shiftmapand Lambda((m)) = {w is an element of {0, 1}(N) : w does not contain 0(m) or 1(m)} for integer m >= 3, studying the Bernoulli-type measures on Lambda((m)) and finding out the unique equivalent sigma-invariant ergodic probability measure.

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