Sequences with constant number of return words
Abstract
An infinite word has the property Rm if every factor has exactly m return words. Vuillon showed that R2 characterizes Sturmian words. We prove that a word satisfies Rm 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 Rm are given by words related to digital expansions in real bases.
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