Analytic order-isomorphisms of countable dense subsets of the unit circle

For functions in Ck(R) which commute with a translation, we prove a theorem on approximation by entire functions which commute with the same translation, with a requirement that the values of the entire function and its derivatives on a specified countable set belong to specified dense sets. Using this theorem, we show that if A and B are countable dense subsets of the unit circle T⊆C with 1∉A , 1∉B , then there is an analytic function h:C∖{0}→C that restricts to an order isomorphism of the arc T∖{1} onto itself and satisfies h(A)=B and h′(z)≠0 when z∈T . This answers a question of P. M. Gauthier.