Non-normal numbers with respect to infinite Markov partitions

In the present paper we investigate two special types of non-normal numbers. On the one hand we call a number extremely non-normal if the set of accumulation points of its frequency of blocks vector is the full set of shift invariant probability vectors. On the other hand we call a number having maximal oscillation frequency if for any fixed block the set of accumulation points of its frequency vector is the full set of possible probability vectors. The goal is to investigate the Baire category of these numbers for infinite Markov partitions.