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# Angular Derivatives at Boundary Fixed Points for Self-Maps of the Disk

Proceedings of the American Mathematical Society
Vol. 126, No. 6 (Jun., 1998), pp. 1697-1708
Stable URL: http://www.jstor.org/stable/118575
Page Count: 12
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## Abstract

Let φ be a one-to-one analytic function of the unit disk D into itself, with φ (0) = 0. The origin is an attracting fixed point for φ , if φ is not a rotation. In addition, there can be fixed points on ∂ D where φ has a finite angular derivative. These boundary fixed points must be repelling (abbreviated b.r.f.p.). The Koenigs function of φ is a one-to-one analytic function σ defined on D such that φ =σ -1(λ σ), where λ =φ ′(0). If φ K is the first iterate of φ that does have b.r.f.p., we compute the Hardy number of σ , h(σ ) = sup{p > 0: σ ∈ Hp( D)}, in terms of the smallest angular derivative of φ K at its b.r.f.p.. In the case when no iterate of φ has b.r.f.p., then $\sigma \in \bigcap _{p<\infty}$Hp, and vice versa. This has applications to composition operators, since σ is a formal eigenfunction of the operator Cφ(f) = f ⚬ φ . When Cφ acts on H2( D), by a result of C. Cowen and B. MacCluer, the spectrum of Cφ is determined by λ and the essential spectral radius of Cφ, re(Cφ). Also, by a result of P. Bourdon and J. Shapiro, and our earlier work, re(Cφ) can be computed in terms of h(σ ). Hence, our result implies that the spectrum of Cφ is determined by the derivative of φ at the fixed point 0 ∈ D and the angular derivatives at b.r.f.p. of φ or some iterate of φ .

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