Growth of powers of non-negative integer matrices In what I am currently doing, there naturally appears the following question: let $A$ be a square matrix with non-negative integer entries. Let $a_n$ be the sum of all entries of $A^n$.
Question: How the sequence $\{a_n\}_{n\geq 1}$ can grow?
Of course if $A$ is positive, then Perron-Frobenius Theorem tells us the answer, but in the general case of non-negative matrices, it can be difficult to guess the asymptotics of the sequence $\{A^n\}_{n\geq 1}$. So, I thought may be there is something known for this case, when we have actually integer matrices. Any references and comments would be appreciated.
 A: I am not sure I understand the question. Any matrix $A$ (integer or not, positive or not) has a Jordan canonical form $A = MJM^{-1},$ whereupon $A^n = M J^n M^{-1}.$ If $A$ is integer and nonsingular, the biggest eigenvalue is at least $1$ in modulus (since the determinant is at least $1$ in absolute value). If it IS equal to $1$ in modulus, the sum of the elements will be polynomial, if it is greater than one, it will be exponential -- if there is a single eigenvalue of maximal modulus, it will be really exponential, otherwise at least there will be a positive density subsequence of $n$ for which it is. Which numbers can occur as eigenvalues of nonnegative integer matrices was answered by Doug Lind in:
Lind, D. A.(1-WA)
The entropies of topological Markov shifts and a related class of algebraic integers. 
Ergodic Theory Dynam. Systems 4 (1984), no. 2, 283–300. 
58F11 (15A48 28D20) 
MR1149738 (92m:11117) Reviewed 
Lind, Douglas(1-WA)
Matrices of Perron numbers. 
J. Number Theory 40 (1992), no. 2, 211–217. 
11R06 (15A48 58F03) 
