Iwip endomorphisms of free groups and fixed points of graph selfmaps

In a paper from 2011, Jiang, Wang and Zhang studied the fixed points and fixed subgroups of selfmaps on a connected finite graph or a connected compact hyperbolic surface . In particular, for any selfmap f : X -> X f\colon X\to X , they proved that a certain quantity defined in terms of the characteristic chr ( f , F ) \operatorname{chr}(f,\mathbf{F}) and the index ind ( f , F ) of a fixed point class of is bounded below by 2 chi ( X ), where chi ( X ) is the Euler characteristic of . In this paper, we give a sufficient condition for when equality holds and hence we partially answer a question of Jiang. We do this by studying iwip outer endomorphisms of free groups acting on stable trees.