Upper bounds for the moduli of polynomial-like maps

TL;DR

This paper extends inequalities relating to polynomial-like maps, showing that bounded moduli imply bounded combinatorics and analyzing parameter slices of cubic polynomials, with implications for complex dynamics.

Contribution

It introduces a new inequality for polynomial-like restrictions of polynomials and applies it to understand parameter spaces and dynamics of cubic polynomials.

Findings

Bounded moduli imply bounded combinatorics for polynomials.

Main cubioid intersects with the closure of the principal hyperbolic domain.

The established inequality generalizes previous results in polynomial dynamics.

Abstract

We establish a version of the Pommerenke-Levin-Yoccoz inequality for the modulus of a polynomial-like restriction of a global polynomial and give two applications. First it is shown that if the modulus of a polynomial-like restriction of an arbitrary polynomial is bounded from below then this forces bounded combinatorics. The second application concerns parameter slices of cubic polynomials given by a non-repelling value of a fixed point multiplier. Namely, the intersection of the main cubioid and the multiplier slice lies in the closure of the principal hyperbolic domain, with only possible exception of queer components.