Let $\alpha,\lambda,\mu$ be nonzero complex numbers with $|\alpha| \gt 1$, and let $u_n=|\lambda \alpha^n + \mu \bar{\alpha}^n|$.
My question: Can anyone determine the set of all so-called adherence values of $(u_n)$, i.e. the values in $(0,+\infty)$ that are the limit of some subsequence of $(u_n)$.
What I tried: Write the polar decomposition of $\alpha$ : $\alpha =r e^{i\theta}$ with $r>0$ and $\theta\in{\mathbb R}$. When $\frac{\theta}{2\pi}$ is rational, it is easy to see that $v_n=\frac{u_n}{r^n}$ is periodic, so that the only possible adherence values for $u_n$ are either $0$ or $+\infty$. I have no ideas of how to proceed when $\frac{\theta}{2\pi}$ is irrational, however.