The statement I am concerned with is this:
Let $\varphi : \mathbb{P}^r_{\mathbb{Z}_p} \to \mathbb{P}^r_{\mathbb{Z}_p}$ be a morphism of degree higher than one. Then the set of $\mathbb{Q}_p$-rational periodic points for $\varphi$ is finite.
The assumption that $\varphi$ is a morphism over $\mathbb{Z}_p$ (and not just over $\mathbb{Q}_p$) means that we deal with an iteration having a good reduction. An example showing the necessity of such an assumption is $r=1$ and $\varphi_{\mathbb{Q}_p}(z) = (z^p-z)/p$: all the periodic points in this example lie in $\mathbb{Z}_p \cup \{\infty\}$. (Of course, the example has no model over $\mathbb{Z}_p$.)
I am aware of the following results in the literature:
When $r = 1$ the statement follows from a more precise result (Theorem 50) in Zieve's thesis Cycles of polynomial mappings (1996), which is also mentioned as Theorem 2.28 in Silverman's book The Arithmetic of Dynamical Systems.
When $\varphi$ is a polynomial in $r$ variables with coefficients from $\mathbb{Z}_p$, the statement holds provided one only considers points in $\mathbb{Z}_p^r$ (even if $\varphi$ is not a morphism of the projective space). This appears in a paper by T. Pezda, Cycles of polynomial mappings in several variables, Manuscripta Math. (1994).
[In both cases, the finite set in question is bounded uniformly in $r, p$, and $\deg{\varphi}$, and in fact the maximum period of a $\mathbb{Q}_p$-point of finite order is bounded in terms of $r$ and $p$ alone. ]
Question. Has the above statement been proved or considered in the literature?
