Hausdorff dimension of frequency sets of univoque sequences

Abstract

We study the set  consisting of univoque sequences, the set consisting of sequences in which the lengths of consecutive zeros and consecutive ones are bounded, and their frequency subsets , ,  and , , consisting of sequences respectively in and  with frequency, lower frequency and upper frequency of zeros equal to some . The Hausdorff dimension of all these sets are obtained by studying the dynamical system where is the shift map and for integer , studying the Bernoulli-type measures on and finding out the unique equivalent -invariant ergodic probability measure.