  The notion of string attractor has been introduced in [Kempa and Prezza,
2018] in the context of Data Compression and it represents a set of positions
of a finite word in which all of its factors can be "attracted". The smallest
size $\gamma^*$ of a string attractor for a finite word is a lower bound for
several repetitiveness measures associated with the most common compression
schemes, including BWT-based and LZ-based compressors. The combinatorial
properties of the measure $\gamma^*$ have been studied in [Mantaci et al.,
2021]. Very recently, a complexity measure, called string attractor profile
function, has been introduced for infinite words, by evaluating $\gamma^*$ on
each prefix. Such a measure has been studied for automatic sequences and
linearly recurrent infinite words [Schaeffer and Shallit, 2021]. In this paper,
we study the relationship between such a complexity measure and other
well-known combinatorial notions related to repetitiveness in the context of
infinite words, such as the factor complexity and the recurrence. Furthermore,
we introduce new string attractor-based complexity measures, in which the
structure and the distribution of positions in a string attractor of the
prefixes of infinite words are considered. We show that such measures provide a
finer classification of some infinite families of words.
