More than 30 years ago R\'enyi [ 1 ] introduced the representations of real numbers with an arbitrary base β > 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 β and ergodic properties of the corresponding β-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 β having complicated symbolic dynamics of their β-shifts.
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Jörg Schmeling (1997) studied this question.