# Good reduction and Shafarevich-type theorems for dynamical systems with portrait level structures

## TL;DR

This paper establishes finiteness results for dynamical systems on the projective line over number fields with specified reduction properties, focusing on maps with portrait level structures and degree constraints.

## Contribution

It proves finiteness of certain classes of dynamical systems with good reduction outside a set, incorporating portrait structures and degree conditions, extending classical reduction theorems.

## Key findings

- Finiteness of PGL_2(R_S)-equivalence classes for specified triples.
- Analysis of degree 2 maps with Y=X and portrait structures.
- Results applicable to dynamical systems with prescribed reduction and portrait data.

## Abstract

Let $K$ be a number field, let $S$ be a finite set of places of $K$, and let $R_S$ be the ring of $S$-integers of $K$. A $K$-morphism $f:\mathbb{P}^1_K\to\mathbb{P}^1_K$ has simple good reduction outside $S$ if it extends to an $R_S$-morphism $\mathbb{P}^1_{R_S}\to\mathbb{P}^1_{R_S}$. A finite Galois invariant subset $X\subset\mathbb{P}^1_K(\bar{K})$ has good reduction outside $S$ if its closure in $\mathbb{P}^1_{R_S}$ is \'etale over $R_S$. We study triples $(f,Y,X)$ with $X=Y\cup f(Y)$. We prove that for a fixed $K$, $S$, and $d$, there are only finitely many $\text{PGL}_2(R_S)$-equivalence classes of triples with $\text{deg}(f)=d$ and $\sum_{P\in Y}e_f(P)\ge2d+1$ and $X$ having good reduction outside $S$. We consider refined questions in which the weighted directed graph structure on $f:Y\to X$ is specified, and we give an exhaustive analysis for degree $2$ maps on $\mathbb{P}^1$ when $Y=X$.