On the set of optimal homeomorphisms for the natural pseudo-distance associated with the Lie group S1

In this dissertation we study the natural pseudo-distance associated with the Lie group S1, a dissimilarity measure on a space of real-valued maps called filtering functions. We focus our attention on the set of the optimal homeomorphisms for the natural pseudo-distance, i.e. the homeomorphisms that represent the best correspondence between two filtering functions. We examine some differential properties that an homeomorphism has to meet in order to be optimal, and we prove the finiteness of the set of the optimal homeomorphisms if the filtering functions are Morse.