A complex sequence with positive values

Question

Let $\lambda_1,\dots,\lambda_d$ be complex numbers that constitute the spectrum of a nonnegative integer matrix, and $P_1,\dots, P_d$ be complex polynoms, such that the sequence $$u_n=\Sigma_{i=1}^d P_i(n)\lambda_i^n$$ takes only real nonnegative values, i.e. $u_n\geq0, \forall n\in\mathbb N$.

I'm interested in the possible behaviours of $u_n$, and more precisely I would like to show the existence of an integer $k$ such that each sequence $(u_{nk+r})_{n\in\mathbb N}$ for $0\leq r\leq k-1$ is asymptotically equivalent to some $a_r n^{b_r}c_r^n$, where $a_r,b_r,c_r$ are real nonnegative constants depending on $r$.

For instance, $b_r=0$ and/or $c_r=1$are interesting cases that can happen.

Any hint would be appreciated, thanks !

What I tried

So I know that for reducible matrices, Perron-Froebenius theorem tells us that there is a real positive eigenvalue $\rho$ of maximal module, and that the other eigenvalues of maximal module are $\rho e^{\frac{ir\pi}{k}}$ with $0<r<k$. It shoud prevent counter-examples like the one from Gerald Edgar, but I'm missing the last argument to conclude. There are two problems in my opinions:

1) here the matrix is not necessarily irreducible,

2) the other eigenvalues (the ones that are not of maximal module) can also play a part.

Here is the most detailed paper I could find giving information about such eigenvalues, but I don't see how to use this information: The spectra of nonnegative integer matrix via formal power series.

Answer 1

No.

Consider the matrix $$\begin{pmatrix} 7 & 8 & 4 & 4 & 0 \\ 0 & 7 & 4 & 4 & 0 \\ 4 & 4 & 7 & 8 & 0 \\ 4 & 4 & 0 & 7 & 0 \\ 0 & 0 & 0 & 0 & 5 \\ \end{pmatrix}$$ The eigenvalues are $3 \pm 4i$, $11 \pm 4 \sqrt{3}$ and $5$. So $$u_n = (3+4i)^n + (3-4i)^n + 3 \cdot 5^n + 0 \cdot (11+4 \sqrt{3})^n + 0 \cdot (11-4 \sqrt{3})^n$$ is of the required form but does not have the desired asymptotics. (Example found using the method of Geoff Robinson's answer here.)

Requiring that all the $P_i$ be nonzero doesn't help. Let $M$ be the above matrix and let $N$ be the $10 \times 10$ matrix with block form $$\begin{pmatrix} 0 & M^2 \\ \mathrm{Id} & 0 \end{pmatrix}.$$ The eigenvalues of $N$ are $\pm3 \pm 4 i$, $\pm 11 \pm 4 \sqrt{3}$ and $\pm 5$.

Now look at $$10000 \cdot \left( (11+4 \sqrt{3})^n + (-11-4 \sqrt{3})^n +(-11+4 \sqrt{3})^n +(11-4 \sqrt{3})^n \right) $$ $$+100 \cdot 5^n + (-5)^n + 2 ((3+4i)^n + (3-4i)^n) + ((-3+4i)^n + (-3-4i)^n)$$

Of course, the exact powers of $10$ don't matter. The point is that $10000$ is large enough to always make things positive for $n$ even. For $n$ odd, the first term vanishes and $100$ is large enough to make the remaining terms positive, but $(3+4i)^n$ is of the same magnitude as $5^n$, so you can't get an asymtotic of the sort you want.