Compact composition operators with symbol a universal covering map

In this paper we study composition operators, Cϕ, acting on the Hardy spaces that have symbol, ϕ, a universal covering map of the disk onto a finitely connected domain of the form D0\{p1,…,pn}, where D0 is simply connected and pi, i=1,…,n, are distinct points in the interior of D0. We consider, in particular, conditions that determine compactness of such operators and demonstrate a link with the Poincare series of the uniformizing Fuchsian group. We show that Cϕ is compact if, and only if ϕ does not have a finite angular derivative at any point of the unit circle, thereby extending the result for univalent and finitely multivalent ϕ.