I have a square matrix $A(p)$ where all of its elements are positive but its spectral radius $\rho(A(p)) \geq 1$. I do not know the exact values of $A(p)$ since it depends the parameter $p$, but I know the formulas for every element of $A(p)$. I have a vector $x(p)$ which depends on $p$ and all of its elements are positive.
I want to study what happens for some limited integer $1\leq k < \infty$, how large $\Vert A^k x\Vert_1$ will become compared with $\Vert x\Vert_1$? Actually, we can form a finite sequence $\{x, Ax, A^2x, \cdots, A^k x\}$, and we know $b + e^T \sum\limits_{i=0}^k A^ix = 1$ where $e \in \mathbb{R}^{n}$ is a all-one vector and $0<b<\infty$.