arXiv:2106.07745v1 [math.NT] 7 Jun 2021 RIGIDITY AND UNLIKELY INTERSECTIONS FOR STABLE p-ADIC DYNAMICAL SYSTEMS ABSOS ALI SHAIKH AND MABUD ALI SARKAR Abstract. Berger asked the question “To what extent the preperiodic points of a p-adic power series determines a stable p-adic dynamical system” ? In this work we have applied the preperiodic points of an invertible p-adic power series in order to determine the corresponding stable p-adic dynamical system. Contents 1. Introduction and motivation 1 2. Some preliminaries on p-adic dynamical system 2 3. Main Results 5 4. Justification based on a Conjecture 9 References 11 1. Introduction and motivation Let K be the finite extension of the p-adic field Q p with ring of integers O K , and the unique maximal ideal m K . We denote the units in O K by O ∗ K . Let ¯ K be the algebraic closure of K and ¯ m K be the integral closure of m K in ¯ K . Let C p be the p-adic completion of ¯ K and denote m Cp = {z ∈ C p ||z| p < 1}. In his paper [Ber2], Berger studied to what extent the torsion points Tors(F ) of a formal group F over O K determine the formal group. He showed that if Tors(F 1 ) ∩ Tors(F 2 ) is infinite then F 1 = F 2 . He further asked the question, if D is a stable p-adic dynamical system, then: “To what extent the preperiodic points Preper(D) determines D ?” In this work, we have answered this question by proving the following theorem: Theorem. If D 1 and D 2 be two stable p-adic dynamical systems over O K such that Z = Preper(D 1 ) ∩ Preper(D 2 ) is infinite, then D 1 = D 2 . 2020 Mathematics Subject Classification. 11S82 (11S31, 11F85, 13J05, 37P35). Key words and phrases. Nonarchimedean dynamical system; preperiodic points; formal group; rigidity; un- likely intersections. 1