Abel universal functions: boundary behaviour and Taylor polynomials

A holomorphic function f on the unit disc \mathbb{D} belongs to the class \mathcal{U}_{A} (\mathbb{D}) of Abel universal functions if the family \{f_{r}: 0\leq r<1\} of its dilates f_{r}(z):=f(rz) is dense in the space of continuous functions on K , for any proper compact subset K of the unit circle. It has been recently shown that \mathcal{U}_{A} (\mathbb{D}) is a dense G_{\delta} subset of the space of holomorphic functions on \mathbb{D} endowed with the topology of local uniform convergence. In this paper, we develop further the theory of universal radial approximation by investigating the boundary behaviour of functions in \mathcal{U}_{A} (\mathbb{D}) (local growth, existence of Picard points and asymptotic values) and the convergence properties of their Taylor polynomials outside \mathbb{D} .