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

Let $\phi$ be a one-to-one analytic function of the unit disk $\mathbb{D}$ into itself, with $\phi(0)=0$. The origin is an attracting fixed point for $\phi$, if $\phi$ is not a rotation. In addition, there can be fixed points on $\partial \mathbb{D}$ where $\phi$ has a finite angular derivative. These boundary fixed points must be repelling (abbreviated b.r.f.p.). The Koenigs function of $\phi$ is a one-to-one analytic function $\sigma$ defined on $\mathbb{D}$ such that $\phi= \sigma ^{-1}(\lambda \sigma )$, where $\lambda =\phi^\prime(0)$. If $\phi _K$ is the first iterate of $\phi$ that does have b.r.f.p., we compute the Hardy number of $\sigma$, $h(\sigma )=\sup\{p>0: \sigma \in H^p(\mathbb{D} )\}$, in terms of the smallest angular derivative of $\phi _K$ at its b.r.f.p.. In the case when no iterate of $\phi$ has b.r.f.p., then $\sigma \in \bigcap _{p<\infty}H^p$, and vice versa. This has applications to composition operators, since $\sigma$ is a formal eigenfunction of the operator $C_\phi (f)=f\circ\phi$. When $C_\phi$ acts on $H^2(\mathbb{D} )$, by a result of C. Cowen and B. MacCluer, the spectrum of $C_\phi$ is determined by $\lambda$ and the essential spectral radius of $C_\phi$, $r_e(C_\phi )$. Also, by a result of P. Bourdon and J. Shapiro, and our earlier work, $r_e(C_\phi )$ can be computed in terms of $h(\sigma )$. Hence, our result implies that the spectrum of $C_\phi$ is determined by the derivative of $\phi$ at the fixed point $0\in \mathbb{D}$ and the angular derivatives at b.r.f.p. of $\phi$ or some iterate of $\phi$.