Dynamics of piecewise linear maps and sets of nonnegative matrices

We consider maps fK(v)=minA∈KAv and gK(v)=maxA∈KAv, where K is a finite set of nonnegative matrices and by “min” and “max” we mean component-wise minimum and maximum. We transfer known results about properties of gK to fK. In particular we show existence of nonnegative generalized eigenvectors of fK, give necessary and sufficient conditions for existence of strictly positive eigenvector of fK, study dynamics of fK on the positive cone. We show the existence and construct matrices A and B, possibly not in K, such that and for any strictly positive vector v.