My REU partner and I are working on a problem involving iterations of quadratic rational maps over an algebraically closed field $K$ that is complete with respect to a non-trivial non-archimedean absolute value $|\cdot|$. We've reduced it to the problem of determining whether $\phi(x) = k(x + x^{-1})$ is post-critically finite for $0 < |k| < 1$ (that is, determining whether the critical points $\pm 1$ are preperiodic). Our conjecture maintains that $\phi$ is not post-critically finite, but this is where we're stuck.

A simple strategy would be to show that some $n$-periodic point $\gamma$ of $\phi$ strictly attracts a critical point of $\phi$. According to a recent paper (Theorem 1.5, p. 4), if $p$ is the residue characteristic of $K$ and either $p=0$ or $p>2$ (assumptions we're willing to make), then a critical point of $\phi$ will be attracted to the cycle containing an $n$-periodic point $\gamma$ of $\phi$ if the multiplier $m(\phi^n,\gamma)$ is strictly between $0$ and $1$.

Unfortunately, the multipliers of the fixed points $\pm\sqrt{k/(1-k)}$ and $\infty$ are $2k - 1$ and $1/k$, respectively, each of which is at least $1$ in absolute value. The periodic points of minimal period $2$, $\pm\sqrt{-k/(1+k)}$, each have a multiplier of $(2k+1)^2$, which is also at least $1$ in absolute value. We're hesitant to continue this method of checking individual periodic points since the calculations become increasingly cumbersome for minimal period greater than $2$. Does anyone have suggestions for how to either: a) show that a periodic point exists with a multiplier in the desired range, or, more broadly, b) show that $\phi$ is(n't) post-critically finite?