Shadowing and Stability of Non-Invertible p-adic Dynamics

The paper establishes (i) shadowing from a family of contracting right inverses that covers Xp, and (ii) topological stability and strong Lipschitz structural stability on Zp under a finite-branch hypothesis plus a mild separation condition, with precise auxiliary lemmas transferring right-inverse structure to perturbations and a conjugacy criterion for shadowing, expansive maps. The candidate solution correctly sketches shadowing via backward iteration using right inverses and obtains expansivity and topological stability in the open/disjoint special case. However, its structural-stability step is flawed: it asserts a reverse semiconjugacy k by merely interchanging f and g without proving that g also has shadowing/expansivity. The paper closes this gap via Lemma 4.1 (right-inverse transfer to g), Claim 2 (g is shadowing), and Lemma 4.3 (yields a conjugacy), none of which are invoked by the model. Moreover, the model makes an inaccurate remark about the need for a uniform contraction bound c; in the p-adic setting all strict contractions have Lipschitz constant ≤ 1/p, so a uniform c<1 is automatic. Hence the paper’s argument is correct and complete, while the model’s is incomplete at the key structural-stability step and leaves continuity/uniqueness details implicit that the paper proves carefully (e.g., Claims 1–3, Lemma 4.3). See Theorem 3.1 and its proof (Claims 1–3) and Lemmas 4.1 and 4.3 in the paper for the missing ingredients .