Decomposition of supercritical linear-fractional branching processes

t is well known that a supercritical single-type Bienaym\ue9-Galton-Watson process can be viewed as a decomposable branching process formed by two subtypes of particles: those having infinite line of descent and those who have finite number of descendants. In this paper we analyze such a decomposition for the linear-fractional Bienaym\ue9-GaltonWatson processes with countably many types. We find explicit expressions for the main characteristics of the reproduction laws for so-called skeleton and doomed particles