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On the Thue-Morse measure. (English) Zbl 0790.11017
The author obtains a precise description of the Thue-Morse measure and of the Fibonacci measure, which result from a precise description of the frequencies of the words occurring in these sequences. He gives more generally the following interesting theorem:
“Let $$\nu$$ be the unique shift invariant measure on a substitution dynamical system generated by a primitive substitution of constant length. Then, there exist $$c_ 2> c_ 1>0$$ such that $c_ 1\leq n \nu([w_ 1\cdots w_ n])\leq c_ 2,$ for all $$n\geq 1$$ and all $$[w_ 1\cdots w_ n]$$ with $$\nu([w_ 1\cdots w_ n])>0$$.”
Note that in the paper under review there is a misprint at the very end of this theorem, note also that $$[w_ 1\cdots w_ n]$$ denotes, as usual, the cylinder of all the sequences having the word $$w_ 1\cdots w_ n$$ as prefix.
The author conjectures moreover that the above result holds true in the case of non-constant length substitutions. The results on the Thue-Morse and Fibonacci sequences have been extended recently by V. Berthé to many other sequences, including in particular all Sturmian sequences, allowing here to give a precise statement for the “conditional block- entropy” of these sequences.

##### MSC:
 11B85 Automata sequences 28D05 Measure-preserving transformations 11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
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