Countable inverse limits of postcritical ω-limit sets of unimodal maps

Let f be a unimodal map of the interval with critical point c. If the orbit of c is not dense then most points in lim←{[0,1], f} have neighborhoods that are homeomorphic with the product of a Cantor set and an open arc. The points without this property are called inhomogeneities, and the set, I, of inhomogeneities is equal to lim←{ω(c), f|}. In this paper we consider the relationship between the limit complexity of w(c) and the limit complexity of I. We show that if w(c) is more complicated than a finite collection of convergent sequences then I can have arbitrarily high limit complexity. We give a complete description of the limit complexity of I for any possible ω(c).