The non-unit conjecture for Misiurewicz parameters

TL;DR

This paper explores whether the difference between two Misiurewicz parameters can be an algebraic unit, extending previous work and providing new cases under certain irreducibility assumptions in complex dynamics.

Contribution

It advances understanding of algebraic units in the context of Misiurewicz parameters, building on prior research and addressing a dynamical analogue of a known number theory problem.

Findings

Many new cases where the difference of two Misiurewicz parameters is not an algebraic unit

Extension of previous results under a widely believed irreducibility assumption

Connections drawn between dynamical parameters and classical special points

Abstract

A Misiurewicz parameter is a complex number $c$ for which the orbit of the critical point $z=0$ under $z^2+c$ is strictly preperiodic. Such parameters play the same role as special points in dynamical moduli spaces that singular moduli (corresponding to CM elliptic curves) play as special points on modular curves. Building on our earlier work, we investigate whether the difference of two Misiurewicz parameters can be an algebraic unit. (The corresponding question for singular moduli was recently answered in the negative by Li.) We answer this dynamical question in many new cases under a widely believed irreducibility assumption.