  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
$\varphi$, computes the morphism on extended intervals and first elements of
the equidistributed sequences associated with fixed points of $\varphi$.
