Analyticity of the Hausdorff dimension and metric structures on Misiurewicz families of polynomials

TL;DR

This paper establishes a natural geometric structure on certain polynomial families, demonstrating the analyticity of the Hausdorff dimension of Julia sets and the metric on parameter space using thermodynamic formalism and spectral gap techniques.

Contribution

It introduces a new dynamically meaningful metric on stable components of polynomial families and proves the analyticity of the Hausdorff dimension of Julia sets within this framework.

Findings

Constructed a natural path metric on parameter space using a 2-form.

Proved the analyticity of the Hausdorff dimension of Julia sets over the parameter space.

Demonstrated the spectral gap property of transfer operators in this context.

Abstract

Consider a holomorphic family $(f_\lambda)_{\lambda \in \Lambda}$ of polynomial maps on $\mathbb C$ with the property that a critical point of $f_\lambda$ is persistently preperiodic to a repelling periodic point of $f_\lambda$. Let $\Omega$ be a bounded stable component of $\Lambda$ with the property that, for all $\lambda \in \Omega$, all the other critical points of $f_\lambda$ belong to attracting basins. In this paper, we introduce a dynamically meaningful geometry on $\Omega$ by constructing a natural path metric on $\Omega$ coming from a 2-form $\langle \cdot, \cdot \rangle_G$. Our construction uses thermodynamic formalism. A key ingredient is the spectral gap of adapted transfer operators on suitable Banach spaces, which also implies the analyticity of $\langle \cdot, \cdot \rangle_G$ on the unit tangent bundle of $\Omega$. As part of our construction, we recover a result of Skorulski and Urba\'nski stating that the Hausdorff dimension of the Julia set of $f_\lambda$ varies analytically over $\Omega$.