Maximal Generic Iterated Galois Images Do Not Determine p-Adic Julia Sets

An AIM problem asks whether the arithmetic or geometric Galois group of the generic iterated-preimage tower of a rational map over a \(p\) -adic field determines its Julia set. We give a uniform negative answer. For every prime \(p\) , the quadratic polynomials \[ z^2+1\qquad\text{and}\qquad z^2-p^{-6} \] over \(\Qp\) have arithmetic and geometric generic iterated Galois images equal to the full group \(\Aut(T_2)\) . The first map has good reduction: its Berkovich Julia set is the Gauss point and its classical Julia set is empty. The second has a type-I Cantor Julia set; moreover, its two contracting inverse branches are defined over \(\Qp\) , so the classical Cantor set already lies in \(\Pone(\Qp)\) . Thus even the full rooted-tree action and the marked normal pair of arithmetic and geometric images fail to determine the cardinality or topological type of the Julia set. The result combines Pink's maximality theorem with a base-field inverse-branch construction. This note is unrefereed and makes no absolute priority claim.