Degree gaps for multipliers and the dynamical Andre-Oort conjecture

TL;DR

This paper extends previous results by showing that polynomial families with infinitely many specializations cannot have low-degree multipliers for periodic cycles, except when these multipliers are zero, advancing understanding in dynamical systems and number theory.

Contribution

It generalizes earlier findings by combining recent work and modifications to demonstrate restrictions on multipliers in polynomial families with infinitely many specializations.

Findings

Polynomial families with infinitely many specializations lack low-degree multipliers for periodic cycles.

Non-zero multipliers in such families must have degree above a certain threshold.

The results extend and unify previous work by Baker, DeMarco, Favre, and Gauthier.

Abstract

We demonstrate how recent work of Favre and Gauthier, together with a modification of a result of the author, shows that a family of polynomials with infinitely many post-critically finite specializations cannot have any periodic cycles with multiplier of very low degree, except those which vanish, generalizing results of Baker and DeMarco, and Favre and Gauthier.