Boundary values via reproducing kernels: The Julia-Carath\'eodory theorem

Given a reproducing kernel $k$ on a nonempty set $X$, we define the reproductive boundary of $X$ with respect to $k$. Furthermore, we generalize the well known nontangential and horocyclic approach regions of the unit circle to this new kind of boundary. We also introduce the concept of a composition factor of $k$, an abstract analogue of analytic selfmaps of the unit disk. Using these notions, we obtain a far reaching generalization of the Julia-Carath\'eodory theorem, stated on an arbitrary set. We also prove Julia's lemma in the abstract setting and give sufficient conditions for the convergence of iterates of some selfmaps. As an application we improve the classical theorem on the unit disk for contractive multipliers of standard weighted Dirichlet spaces, as well as Besov spaces on the unit ball. Many examples and questions are provided for these novel objects of study.