## Substitution invariant Sturmian bisequences.(English)Zbl 0978.11005

A doubly infinite word over the two-element alphabet $$\{0,1\}$$ is called Sturmian if on its position we have either $$\lfloor (n+1+k)\alpha+\rho\rfloor-\lfloor (n+k)\alpha+\rho\rfloor- \lfloor \alpha\rfloor$$ or $$\lceil (n+1+k)\alpha+\rho\rceil-\lceil (n+k)\alpha+\rho\rceil- \lceil\alpha\rceil$$ for each $$n\in{\mathbb{Z}}$$ and some $$k\in{\mathbb{Z}}$$, where $$\alpha$$ is irrational and $$\rho$$ real. A right-sided infinite word $$y$$ is Sturmian if there exists a left-sided infinite word $$y^\prime$$ such that $$y^\prime y$$ is Sturmian. A substitution $$f$$ (i.e. a map over the free monoid over $$\{0,1\}$$ preserving concatenation) is Sturmian if $$f(w)$$ is a right-sided infinite Sturmian whenever $$w$$ is. The author proves a condition on $$\alpha$$ and $$\rho$$ equivalent to the fact that a Sturmian bisequence is fixed (up to a shift of the indices of its terms) by a Sturmian substitution.

### MSC:

 11B83 Special sequences and polynomials 11B85 Automata sequences 68R15 Combinatorics on words
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### References:

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