Morphic words, Beatty sequences and integer images of the Fibonacci language

Abstract

Morphic words are letter-to-letter images of fixed points x of morphisms on finite alphabets. There are situations where these letter-to-letter maps do not occur naturally, but have to be replaced by a morphism. We call this a decoration of x. Theoretically, decorations of morphic words are again morphic words, but in several problems the idea of decorating the fixed point of a morphism is useful. We present two of such problems. The first considers the so called AA sequences, where α is a quadratic irrational, A is the Beatty sequence defined by A(n)=⌊αn⌋, and AA is the sequence (A(A(n))). The second example considers homomorphic embeddings of the Fibonacci language into the integers, which turns out to lead to generalized Beatty sequences with terms of the form V(n)=p⌊αn⌋+qn+r, where p,q and r are integers.