More than 30 years ago R\'enyi [1] introduced the representations of real numbers with an arbitrary base $\beta > 1$ as a generalization of the $p$-adic representations. One of the most studied problems in this field is the link between expansions to base $\beta$ and ergodic properties of the corresponding $\beta$-shift.

In this paper we will follow the bibliography of Blanchard [2] and give an affirmative answer to a question on the size of the set of real numbers $\beta$ having complicated symbolic dynamics of their $\beta$-shifts.