Burrows-Wheeler Transform on Purely Morphic Words

The study of the compressibility of repetitive sequences is an issue that is attracting great interest. We consider purely morphic words, which are highly repetitive sequences generated by iterating a morphism
$\varphi$
that admits a fixed point (denoted by
$\varphi^{\infty}(a)$
) starting from a given character
$a$
belonging to the finite alphabet
$A$
, i.e.
$\varphi^{\infty}(a)=\lim\nolimits_{i\rightarrow\infty}\varphi^{i}(a)$
. Such morphisms are called prolongable on
$a$
. Here we focus on the compressibility via the Burrows-Wheeler Transform (
$BWT$
) of infinite families of finite sequences generated by morphisms. In particular, denoted by
$r(w)$
the number of equal-letter runs of a word
$w$
, we provide new upper bounds on
$r(\mathsf{bwt} (\varphi^{i}(a)))$
, i.e. the number of equal-letter runs produced when
$BWT$
is applied on
$\varphi^{i}(a)$
. Such bounds depend on the factor complexity
$f_{x}(n)$
of the infinite word
$x=\varphi^{\infty}(a)$
, that counts, for each
$n\geq 0$
, the number of distinct factors of
$x$
having length
$n$
.