Consider a set of positive matrices $(P_k)_{k\in K}$ in $\mathbb R^{p\times p}_{++}$ ($P_k$ is positive in the sense that all entries of $P_k$ are positive).

Let $X_0\in\mathbb R_{++}^p$ and define for every $n\ge0$ $X_{n+1}=\max_{k\in K}\{ P_kX_n\}$, where the max operator has to be understood element-wise: for all $i\in\{1,\ldots,p\}$, $[X_{n+1}]_i=\max\{k\in K:[P_kX_n]_i\}$.

Do the following statements hold?

- There exists a positive matrix $P$ and $n_0\ge0$ such that for all $n\ge n_0$, $X_{n+1}=PX_n$.
- The matrix $P$ is constructed as follows. Consider the set of matrices, such that every row $i\in\{1,\ldots,p\}$ of a matrix $M$ in that set is the row $i$ of a matrix $P_k$. (There are $(Card\ K)^p$ such matrices.) The matrix $P$ is the element of this set, which has the largest Perron-Frobenius eigenvalue, denoted $\lambda_{PF}$.
- $(X_n)_n$ converges to the Perron-Frobenius eigenvector. More precisely, $\lim_{n\rightarrow\infty} X_n/||X_n||$ exists and is denoted $x$. It is such that: $Px = \lambda_{PF} x$.

I inferred the previous properties from the case $p=2$ but even in that simple case, I do manage to prove any of the above statements.

Thanks for any help or useful reference.