Morphic words and equidistributed sequences

Abstract

The problem we consider is the following: Given an infinite word w on an ordered alphabet, construct the sequence νw=(ν[n])n, equidistributed on [0,1] and such that ν[m]<ν[n] if and only if σm(w)<σn(w), where σ is the shift operation, erasing the first symbol of w. The sequence νw 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 νw 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 νw in this case is also constructed with a morphism.At last, we introduce a software tool which, given a binary morphism φ, computes the morphism on extended intervals and first elements of the equidistributed sequences associated with fixed points of φ.