Conditions for the irreducibility and primitivity of monotone subhomogeneous mappings

We present necessary and sufficient conditions for the local irreducibility of monotone subhomogeneous transformations of the cone $\mathbb{R}_+^q$. The main attention is paid to the notion of irreducibility of a mapping at zero, which is a weakening of the classical notion of irreducibility of a mapping. We analyze the properties of monotone first-degree positively homogeneous mappings irreducible at zero and of subhomogeneous mappings. Necessary and sufficient conditions are obtained for the primitivity of such mappings.