Countable inverse limits of postcritical \omega-limit sets of unimodal maps

Abstract

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 {{\omega}(c),f| {\omega}(c) }. In this paper we consider the relationship between the limit complexity of {\omega}(c) and the limit complexity of I. We show that if {\omega}(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 {\omega}(c).