Non-normal numbers in dynamical systems fulfilling the specification property

In the present paper we want to focus on this dichotomy of the non-normal numbers – on the one hand they are a set of measure zero and on the other hand they are residual – for dynamical system fulfilling the specification property. These dynamical systems are motivated by $\beta$-expansions. We consider the limiting frequencies of digits in the words of the languagse arising from these dynamical systems, and show that not only a typical $x$ in the sense of Baire is non-normal, but also its Ces`aro variants diverge.