Dynamics of piecewise linear maps and sets of nonnegative matrices

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 $v$.Comment: 20 page

On Sushchansky p-groups

We study Sushchansky p-groups. We recall the original definition and translate it into the language of automata groups. The original actions of Sushchansky groups on p-ary tree are not level-transitive and we describe their orbit trees. This allows us to simplify the definition and prove that these groups admit faithful level-transitive actions on the same tree. Certain branch structures in their self-similar closures are established. We provide the connection with, so-called, G groups that shows that all Sushchansky groups have intermediate growth and allows to obtain an upper bound on their period growth functions.Comment: 14 pages, 3 figure