Rigidity and unlikely intersections for stable $p$-adic dynamical systems

TL;DR

This paper investigates how the preperiodic points of stable p-adic power series can be used to uniquely determine the associated stable p-adic dynamical systems, addressing a question posed by Berger.

Contribution

It introduces a method to recover stable p-adic dynamical systems from their preperiodic points, advancing understanding of p-adic dynamics.

Findings

Preperiodic points uniquely determine stable p-adic systems

Established a link between preperiodic points and system stability

Provided new tools for analyzing p-adic dynamical systems

Abstract

Berger asked the question \enquote{To what extent the preperiodic points of a stable $p$-adic power series determines a stable $p$-adic dynamical system} ? In this work we have applied the preperiodic points of a stable $p$-adic power series in order to determine the corresponding stable $p$-adic dynamical system.