Angular derivatives at boundary fixed points for self-maps of the disk

Abstract

Let X be a one-to-one analytic function of the unit disk D into itself, with O(O) = 0. The origin is an attracting fixed point for X, if X is not a rotation. In addition, there can be fixed points on &19D where X has a finite angular derivative. These boundary fixed points must be repelling (abbreviated b.r.f.p.). The Koenigs function of X is a one-to-one analytic function a defined on D0 such that X = -'(Ao}), where A = O'(0). If OK is the first iterate of / that does have b.r.f.p., we compute the Hardy number of a, h(a) = sup{p > 0: a E HP(D)}, in terms of the smallest angular derivative of OK at its b.r.f.p.. In the case when no iterate of X has b.r.f.p., then a E np . When COk acts on H2(D), by a result of C. Cowen and B. MacCluer, the spectrum of C0, is determined by A and the essential spectral radius of CQ, re(C4s). Also, by a result of P. Bourdon and J. Shapiro, and our earlier work, re (CO) can be computed in terms of h(o). Hence, our result implies that the spectrum of CQ is determined by the derivative of X at the fixed point 0 E D and the angular derivatives at b.r.f.p. of X or some iterate of k.