Pressure path metrics on parabolic families of polynomials

TL;DR

This paper constructs a pressure metric on certain families of complex polynomials, providing a new geometric tool analogous to metrics used in Hitchin components, with implications for understanding polynomial dynamics.

Contribution

It introduces a pressure form on subfamilies of polynomial moduli spaces defined by parabolic relations, establishing a new path metric in complex dynamics.

Findings

Constructed a positive semi-definite pressure form on the family

Proved the pressure form defines a path metric on the parameter space

Draws an analogy with pressure metrics in Hitchin components

Abstract

Let $\Lambda$ be a subfamily of the moduli space of degree $D\ge2$ polynomials defined by a finite number of parabolic relations. Let $\Omega$ be a bounded stable component of $\Lambda$ with the property that all critical points are attracted by either the persistent parabolic cycles or by attracting cycles in $\mathbb C$. We construct a positive semi-definite pressure form on $\Omega$ and show that it defines a path metric on $\Omega$. This provides a counterpart in complex dynamics of the pressure metric on cusped Hitchin components recently studied by Kao and Bray-Canary-Kao-Martone.