Cones for coordinate-wise monotone functions and invariant manifolds of monotone maps

In this paper we give a characterization of monotone discrete systems of equations in terms of associated signature matrix and give some properties of certain invariant surfaces of codimension 1, which often give the boundary of attraction of some fixed points. We present several examples that illustrate our results in the case of k-dimensional systems where $ k \geq 3 $ k≥3.