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Substitution invariant cutting sequences. (English) Zbl 0786.11041
Let $$\alpha$$ be a real number. The sequence $$f_ \alpha$$ of 0’s and 1’s arising from the formula $$f_ \alpha(n)= [(n+1)\alpha]- [n\alpha]$$ can be transformed by substituting the 0’s and 1’s by finite strings of 0’s and 1’s respectively. A complete characterisation of those $$\alpha$$ for which there is an invariant non-trivial substitution is obtained by relating the $$f_ \alpha$$ to cutting sequences and continued fraction expansions. For example, if $$\alpha$$ is an irrational in $$(0,{1\over 2})$$, then $$f_ \alpha$$ is invariant under some non-trivial substitution if and only if $$\alpha=[0,a_ 1,\overline{a_ 2,\dots, a_ n}]$$, where $$a_ n+1\geq a_ 1\geq 2$$.

##### MSC:
 11J70 Continued fractions and generalizations 11B99 Sequences and sets 11A55 Continued fractions
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