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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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