On the Thue-Morse measure.

*(English)*Zbl 0790.11017The 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. BerthĂ© 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.

“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. BerthĂ© 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.

Reviewer: J.-P.Allouche (Marseille)

##### MSC:

11B85 | Automata sequences |

28D05 | Measure-preserving transformations |

11K55 | Metric theory of other algorithms and expansions; measure and Hausdorff dimension |