A tree automaton with infinitely many rogue fixed points under every supercritical offspring law

Johnson, Podder and Skerman asked whether a tree automaton with finitely many, but at least three, states can have infinitely many fixed points of its automaton distributional map. Under binary branching an example with three states is implicit in earlier work of Rozikov and Zada on quadratic stochastic operators. We show that the phenomenon holds for every supercritical offspring law, Poisson laws included. For every $k\ge2$ there is an automaton on $k+1$ states whose fixed points, for an offspring law with generating function $G$ and mean greater than one, are one point mass together with the $(k-1)$-dimensional simplex of laws that give the distinguished state mass $s_*$, where $G'(s_*)=1$. Every point of the simplex is rogue, with no moment assumption on the offspring law; for the points other than the $k$ vertices this has a short self-contained proof.