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

If $\varphi$ and $\psi$ 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 $d_G(\varphi,\psi)$ is defined as the infimum of $\mathcal{L}(g)=\|\varphi-\psi \circ g \|_\infty$, 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 $X=G=S^1$. In particular, we study the set of the optimal homeomorphisms for $d_G$, i.e. the elements $\rho_\alpha$ of $S^1$ such that $\mathcal{L}(\rho_\alpha)$ is equal to $d_G(\varphi,\psi)$. 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.