Dynamics of piecewise linear maps and sets of nonnegative matrices

Abstract

We consider functions $f(v)={\min }_{A\in K}Av$ and $g(v)={\max }_{A\in K}Av$, where $K$ is a finite set of nonnegative matrices and by "min" and "max" we mean coordinate-wise minimum and maximum. We transfer known results about properties of $g$ to $f$. In particular we show existence of nonnegative generalized eigenvectors for $f$, give necessary and sufficient conditions for existence of strictly positive eigenvector for $f$, study dynamics of $f$ on the positive cone. We show the existence and construct matrices $A$ and $B$, possibly not in $K$, such that $f^n(v)\sim A^{n}v$ and $g^n(v)\sim B^{n}v$ for any strictly positive vector $v$.

20 pages