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Jan 7, 2015 at 9:50 comment added naf I don't know whether the statement holds over $\mathbb{F}_q[[T]]$. As an aside, the finiteness result (over $\mathbb{Q}_p$) can actually be extended to preperiodic points using a compactness argument, but the bound in this case actually depends on $\varphi$.
Jan 7, 2015 at 8:03 comment added Vesselin Dimitrov @ulrich: Thank you! Do you know if the analogous statement continues to hold over $\mathbb{F}_q[[T]]$? This appears to be left open by Fakhruddin's paper (cf. Theorem 1 (ii) and the Remark following it).
Jan 7, 2015 at 6:39 comment added naf This is true. See, for example, arxiv.org/abs/math/0212200.
Jan 7, 2015 at 0:34 history asked Vesselin Dimitrov CC BY-SA 3.0