S-integral preperiodic points for dynamical systems over number fields

TL;DR

This paper proves a finiteness result for S-integral preperiodic points of rational maps over number fields, under certain local conditions, advancing understanding of arithmetic dynamics.

Contribution

It verifies a special case of S. Ih's conjecture by establishing finiteness of S-integral preperiodic points with respect to a non-preperiodic point under specific local conditions.

Findings

Finiteness of S-integral preperiodic points under given conditions

Verification of a special case of S. Ih's conjecture

Conditions involving local properties at each place

Abstract

Given a rational map $\phi: {\mathbb P}^1\to {\mathbb P}^1$ defined over a number field $K$, we prove a finiteness result for $\phi$-preperiodic points which are $S$-integral with respect to a non-preperiodic point $P$, provided $P$ satisfies a certain local condition at each place. This verifies a special case of a conjecture of S. Ih.