Fixed points of infinite-dimensional generating functions

This paper is concerned with the characterizations of fixed points of the generating function of branching processes with countably many types. We assume each particle of type $i$ can only give offspring of type $j\geq i$, whose number only depend on $j-i$. We prove that, for these processes, there are at least countably many fixed points of the offspring generating function, while the extinction probability set of the process has only $2$ elements. This phenomenon contrasts sharply with those of finite-type branching processes. Our result takes one step forward on the related conjecture on the fixed points of infinite-dimensional generating functions in literatures