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

## TL;DR

This paper investigates the topological structure of inhomogeneities in the inverse limit spaces of unimodal maps, revealing how the complexity of the critical orbit influences the complexity of the inhomogeneity set.

## Contribution

It provides a complete classification of the limit complexity of the inhomogeneity set based on the complexity of the critical orbit's omega-limit set.

## Key findings

- Inhomogeneity set can have arbitrarily high limit complexity.
- Complete description of the relationship between omega-limit set complexity and inhomogeneity complexity.
- If the critical orbit's omega-limit set is complex, the inhomogeneity set can be highly complex.

## 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).