A Dynamical Néron--Ogg--Shafarevich Criterion via Orbital Arboreal Representations

Let $K$ be a non-archimedean local field and $\phi :{\mathbb{P}}^1\to {\mathbb{P}}^1$ a rational endomorphism of degree $d\ge 2$ over $K$. In the tame case ($p\nmid d$), we show that strict good reduction is equivalent to the existence of a nonempty Zariski open subset $U_k\subset {\mathbb{P}}_k^1\setminus \mathrm{P}\mathrm{C}(\tilde{\phi })$ over which the canonical residual morphism is finite étale of degree $d$. The criterion separates two complementary local invariants of a normalized integral lift: $\mathrm{R}\mathrm{e}\mathrm{s}(F,G)$ controls residual degree drop, while the fiber discriminants $\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{c}(F_{n,x})$ control étaleness of the residual fibers once full residual degree is ensured. Consequently, for every finite $x\in {\mathcal{O}}_K$ with $\bar{x}\in U_k$, the extensions $K(X_n(x))\mathrm{/}K$ are unramified for all $n\ge 1$. We introduce the orbital preimage tree $T_{O^{+}(x)}={\mathrm{}}_n{X}_{\mathrm{\infty }}({\phi }^n(x))$, the colimit in $G_K$-sets along the forward orbit, and the orbital arboreal Galois image ${\mathcal{G}}_{O^{+}(x)}=\mathrm{I}\mathrm{m}(G_K\to \mathrm{A}\mathrm{u}\mathrm{t}(T_{O^{+}(x)}))$. On the forward-invariant safe locus $U_k^{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}}={\bigcap }_{m\ge 0}{\tilde{\phi }}^{-m}(U_k)$, strict good reduction is captured by the bijectivity of the orbital reduction map $X_n(x_m)\to {\tilde{X}}_n({\bar{x}}_m)$. This canonical orbit-invariant framework connects with arboreal Galois representations (Boston-Jones, Jones, and others) and yields pointwise and orbit-level reformulations. Explicit examples over ${\mathbb{Q}}_p$ illustrate the criterion.