A Riemannian metric on polynomial hyperbolic components

TL;DR

This paper introduces a Riemannian metric on hyperbolic components of polynomial moduli spaces based on measure-theoretic entropy, revealing properties of the Hausdorff dimension function and its critical points.

Contribution

It constructs a new Riemannian metric on polynomial hyperbolic components using entropy, and analyzes the Hausdorff dimension function's critical points and maxima.

Findings

Hausdorff dimension function has no local maximum on hyperbolic components

Provided a condition for points not being critical points of the dimension function

Established a metric linking entropy and geometric properties of polynomials

Abstract

We introduce a Riemannian metric on certain hyperbolic components in the moduli space of degree $d \ge 2$ polynomials. Our metric is constructed by considering the measure-theoretic entropy of a polynomial with respect to some equilibrium state. As applications, we show that the Hausdorff dimension function has no local maximum on such hyperbolic components. We also give a sufficient condition for a point not being a critical point of the Hausdorff dimension function.