SUBSTITUTIONS BY MEMORANDUM TO DIMENSION 1

Pattern subsitutions in dimension 1. A substitution is a morphism of the free monoid: each letter is mapped to a word, and the image of a word is the concatenation of the images of its letters. This paper introduces a generalization of the notion of substitution, where the image of a letter is not a word but a pattern, i.e., a 'word with holes': the image of a word is obtained by connecting the patterns corresponding to each of the letters by means of local rules. We completely characterize pattern substitutions which are defined on every biinfinite sequence, and we explain how to build them. We show that every biinfinite sequence which is a fixed point of a pattern substitution is substitutive, i.e., it is the image, by a letter to letter morphism, of a fixed point of a substitution (in the usual meaning).