Cycle Lengths in $A^k b

Let $A$ be a nonnegative, $n \times n$ matrix, and let $b$ be a nonnegative, $n \times n$ vector. Let $S$ be the sequence $\{ A^k b \},k = 0,1,2, \cdots $. Define $m( A,b )$ to be the length of the cycle of zero-nonzero patterns into which $S$ eventually falls. Define $m( A )$ to be the maximum, over all nonnegative $b$ of $m( A,b )$. Finally, define $m( n )$ to be the maximum, over all nonnegative, $n \times n$ matrices $A$ of $m( A )$. This paper shows given $A$ and $b$, that $m( A,b )$ is a divisor of a certain number, which is determined by the structure of $A$ and $b$. It is also shown that $\log m ( n ) \sim ( n\log n )^{1/ 2} $.