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Sequences with constant number of return words

Lubomira Balkova·Edita Pelantova·Wolfgang Steiner
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Abstract

An infinite word has the property $R_m$ if every factor has exactly $m$ return words. Vuillon showed that $R_2$ characterizes Sturmian words. We prove that a word satisfies $R_m$ if its complexity function is $(m-1)n+1$ and if it contains no weak bispecial factor. These conditions are necessary for $m=3$, whereas for $m=4$ the complexity function need not be $3n+1$. New examples of words satisfying $R_m$ are given by words related to digital expansions in real bases.

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Lubomira Balkova, Edita Pelantova, Wolfgang Steiner. 2007-09-27. Sequences with constant number of return words. https://arxiv.org/abs/math/0608603

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