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Symbolic dynamics for $$\beta$$-shifts and self-normal numbers. (English) Zbl 0908.58017
Consider the family of one-dimensional mappings defined in the interval $$[0,1)$$, indexed by $$\beta$$ belonging to the set of real numbers greater than 1, $$T_\beta: x \mapsto \beta x$$. Rényi in 1957 discussed this family as a model for expanding a real number in a noninteger base $$\beta$$. The author writes: “This gave rise to number-theoretic questions – dependence of the expansions on number theoretic properties of $$\beta$$. The second kind of problem is to analyze the symbolic dynamics associated to the transformation $$T_\beta$$. Here we are at the link between coding and automata theory. Finally, the maps $$T_\beta$$ are typical examples of monotone one-dimensional expanding dynamical systems. For this class of systems it is well known that the dynamics are determined by the orbits of critical points. In the case of $$T_\beta$$ this means that all information of the symbolic dynamics is already contained in the expansion of 1.”
The paper deals with questions concerning the size of sets of numbers $$\beta$$ having some (rather complicated) symbolic dynamics of their $$\beta$$-shifts. More precisely, it is proved that the set of $$\beta$$ such that the orbit of 1 (the behaviour of which one can investigate – equivalently – instead of the expansion of 1) is infinite but not dense, has Hausdorff dimension 1 and Lebesgue measure zero. The set of those $$\beta$$ which have dense orbits of 1 is residual in the set of real numbers greater than 1. The author introduces a class of numbers called self-normal numbers defined by means of some properties of their own expansion in their base and shows that almost every number (with respect to the Lebesgue measure) is self-normal, but generically a number is not self-normal.

##### MSC:
 37E05 Dynamical systems involving maps of the interval 37B10 Symbolic dynamics 11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension 11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. 11A67 Other number representations
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