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arXiv · 1807.08321

Morphic words and equidistributed sequences

Mélodie Andrieu·Anna E. Frid
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Abstract

The problem we consider is the following: Given an infinite word $w$ on an ordered alphabet, construct the sequence $\nu_w=(\nu[n])_n$, equidistributed on $[0,1]$ and such that $\nu[m]<\nu[n]$ if and only if $\sigma^m(w)<\sigma^n(w)$, where $\sigma$ is the shift operation, erasing the first symbol of $w$. The sequence $\nu_w$ exists and is unique for every word with well-defined positive uniform frequencies of every factor, or, in dynamical terms, for every element of a uniquely ergodic subshift. In this paper we describe the construction of $\nu_w$ for the case when the subshift of $w$ is generated by a morphism of a special kind; then we overcome some technical difficulties to extend the result to all binary morphisms. The sequence $\nu_w$ in this case is also constructed with a morphism. At last, we introduce a software tool which, given a binary morphism $\varphi$, computes the morphism on extended intervals and first elements of the equidistributed sequences associated with fixed points of $\varphi$.

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Mélodie Andrieu, Anna E. Frid. 2018-07-22. Morphic words and equidistributed sequences. https://arxiv.org/abs/1807.08321

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