A partial classification of inverse limit spaces of tent maps with periodic critical points

We work within the one parameter family of symmetric tent maps, where the slope is the parameter. Given two such tent maps fa, fb with periodic turning points of the same period, we use the finite kneading sequences of the maps to obtain a necessary condition for the inverse limit spaces (I, fa) and (I, fb) to be homeomorphic. As this condition depends only on the parity of the kneading sequence, it is easily checked. To obtain our result, we define topological substructures of a composant, called "wrapping points" and "gaps", and identify properties of these substructures preserved under a homeomorphism. It is known that if the periods differ, then the inverse limit spaces are not homeomorphic.