ORBITS OF AUTOMORPHISM GROUPS OF AFFINE SURFACES OVER p-ADIC FIELDS

The candidate’s outline matches the paper’s Theorem A, including: (i) finiteness of finite Γ-orbits (Step 1 / Theorem D) and the argument via intersections of periodic loci plus the fact that a loxodromic automorphism preserves no curve, (ii) construction of local analytic flows after passing to a finite-index subgroup Γ0 (Theorem 2.5) and use of the Bell–Poonen interpolation, (iii) the algebraic line distribution and the finiteness of the tangency locus T_Γ (Lemma 6.1), and (iv) the finiteness/“clopen” structure of orbit-closures via Theorem B (and its countable variant), together with uniqueness of stationary measures on each orbit-closure via the compact-group closure argument (Proposition 7.2). All of these steps are the engine of the paper’s proof and are invoked in essentially the same way by the candidate. See Theorem A and its items (1)–(4) in the Introduction, Theorem B in §2.6, Theorem 2.5 in §2.4.2, Step 1 in §6 (Theorem D), the set S and T_Γ and Lemma 6.1 in §6, and Proposition 7.2 in §7 (compact-group uniqueness) .