The structure of the fixed point set of a reducible monotone subhomogeneous mapping

We analyze the structure of the set of nontrivial equilibria for a monotone subhomogeneous discrete-time dynamical system on the nonnegative orthant of a finite-dimensional Euclidean space under as weak additional assumptions as possible. We use the notion of local irreducibility of a nonlinear mapping introduced by the authors. It is shown that, if a monotone subhomogeneous mapping has positive fixed points lying on different rays starting at the origin, then this mapping is reducible at at least one of them and a part of the components of the mapping are positively homogeneous on segments of these rays containing the positive fixed points. In particular, for concave mappings, this means the reducibility of the mapping at zero. As a result, we obtain a generalization of the theorem on the uniqueness of the ray containing the positive fixed points of such a mapping with the only additional assumption that the mapping is irreducible on the set of its positive fixed points. In this case, the set of all positive fixed points of a monotone subhomogeneous mapping forms a continuous part of some ray starting at the origin.