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

If φ and ψ are two continuous real-valued functions defined on a compact topological space X and G is a subgroup of the group of all homeomorphisms of X onto itself, the natural pseudo-distance dG(φ,ψ) is defined as the infimum of L(g)=‖φ−ψ∘g‖∞, as g varies in G. In this paper, we make a first step towards extending the study of this concept to the case of Lie groups, by assuming X=G=S1. In particular, we study the set of the optimal homeomorphisms for dG, i.e. the elements ρα of S1 such that L(ρα) is equal to dG(φ,ψ). As our main results, we give conditions that a homeomorphism has to meet in order to be optimal, and we prove that the set of the optimal homeomorphisms is finite under suitable conditions.