Asymptotic values and Baire category

Abstract:

Let $f$ be meromorphic in the unit disc, and let $\alpha$ be a complex number. Given $\varepsilon > 0$, let ${T_\varepsilon }(\alpha )$ denote the set of points ${e^{i\theta }}$ for which the cluster set ${C_\mathcal {L}}(f,{e^{i\theta }})$ lies in the $\varepsilon$-neighbourhood of $\alpha$ for some arc $\mathcal {L} \to {e^{i\theta }}$. Then a sufficient condition that the set of points on the unit circle at which $f$ possesses point-asymptotic value $\alpha$ be of first category is that ${T_\varepsilon }(\alpha )$ contains no arc for some $\varepsilon > 0$.