Misiurewicz parameters and dynamical stability of polynomial-like maps of large topological degree

TL;DR

This paper explores the relationship between Misiurewicz parameters and the stability of polynomial-like maps with large topological degree, extending existing bifurcation theory to this broader setting.

Contribution

It generalizes the stability and bifurcation theory for endomorphisms of projective space to polynomial-like maps of large topological degree, linking Misiurewicz parameters to postcritical volume growth.

Findings

Misiurewicz parameters are characterized by growth conditions of postcritical volume.

The theory of stability and bifurcation is extended to polynomial-like maps.

A connection between dynamical stability and topological degree is established.

Abstract

Given a family of polynomial-like maps of large topological degree, we relate the presence of Misiurewicz parameters to a growth condition of the postcritical volume. This allows us to generalize to this setting the theory of stability and bifurcation developed by Berteloot, Dupont and the author for endomorphisms of $\mathbb{P}^k$.