Hausdorff dimension of particularly non-normal numbers in dynamical systems fulfilling the specification property

In this paper, we consider non-normal numbers occurring in dynamical systems fulfilling the specification property. It has been shown that in this case the set of non-normal numbers has measure zero. In the present papers we show that a smaller set, namely the set of particularly non-normal numbers, has full Hausdorff dimension. A particularly non-normal number is a number $x$ such that there exist two digits, one whose limiting frequency in $x$ exists and another one whose limiting frequency in $x$ does not exist.