Dynamics of piecewise linear maps and sets of nonnegative matrices

Details Dynamics of piecewise linear maps and sets of nonnegative matrices Dynamics of piecewise linear maps and sets of nonnegative matrices

2006-06-02

arxiv.org/abs/math/0606047v2

Mathematics

Optimization and Control

20 pages

Scientific paper

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 $f^n(v)\sim A^nv$ and $g^n(v)\sim B^nv$ for any strictly positive vector $v$.

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