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 ${\nu }_w=(\nu [n]{)}_n$, equidistributed on $[0,1]$ and such that $\nu [m]<\nu [n]$ if and only if ${\sigma }^m(w)<{\sigma }^n(w)$, where $\sigma$ is the shift operation, erasing the first symbol of $w$. The sequence ${\nu }_w$ 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 ${\nu }_w$ 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 ${\nu }_w$ in this case is also constructed with a morphism. At last, we introduce a software tool which, given a binary morphism $\phi$, computes the morphism on extended intervals and first elements of the equidistributed sequences associated with fixed points of $\phi$.

References in corpus (4)

• The number of permutations realized by a shift
• An Algorithm Enumerating All Infinite Repetitions in a D0L System
• Synchronizing delay for binary uniform morphisms
• Canonical Representatives of Morphic Permutations