A partial classification of inverse limits with irreducible functions

We discuss a class of upper semi-continuous set-valued functions called irreducible functions, and we develop multiple tools for distinguishing topologically between their inverse limits. First, we discuss properties of subcontinua of their inverse limits, and we use those properties to show that for certain irreducible functions, given two, if their graphs contain different (finite) numbers of maximal nowhere dense arcs, then they have topologically distinct inverse limits. Additionally, we discuss endpoints of inverse limits with irreducible functions, and finally, we apply these tools to obtain a complete classification of the inverse limits arising from four specific families of irreducible functions.