# Critical orbits of polynomials with a periodic point of specified multiplier

## TL;DR

This paper investigates the structure of polynomial and rational maps with specific periodic points, demonstrating boundedness properties in moduli spaces and extending results to function fields and quadratic rational maps.

## Contribution

It establishes bounded height results for polynomials with a parabolic fixed point and limited critical orbits, answering a question by Adam Epstein and extending to rational maps.

## Key findings

- Bounded height set of conjugacy classes with parabolic fixed points
- Extension of results to function fields
- Analogous boundedness result for quadratic rational maps

## Abstract

Answering a question posed by Adam Epstein, we show that the collection of conjugacy classes of polynomials admitting a parabolic fixed point and at most one infinite critical orbit is a set of bounded height in the relevant moduli space. We also apply the methods over function fields to draw conclusions about algebraically parametrized families, and prove an analogous result for quadratic rational maps.