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

Abstract

Consider a holomorphic family (fλ)λ∈Λ$(f_\lambda)_{\lambda \in \Lambda }$ of polynomial maps on C$\mathbb {C}$ with the property that a critical point of fλ$f_\lambda$ is persistently preperiodic to a repelling periodic point of fλ$f_\lambda$ . Let Ω$\Omega$ be a bounded stable component of Λ$\Lambda$ with the property that, for all λ∈Ω$\lambda \in \Omega$ , all the other critical points of fλ$f_\lambda$ belong to attracting basins. In this paper, we introduce a dynamically meaningful geometry on Ω$\Omega$ by constructing a natural path metric on Ω$\Omega$ coming from a 2‐form ⟨·,·⟩G$\langle \cdot, \cdot \rangle _G$ . 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 ⟨·,·⟩G$\langle \cdot, \cdot \rangle _G$ on the unit tangent bundle of Ω$\Omega$ . As part of our construction, we recover a result of Skorulski and Urbański stating that the Hausdorff dimension of the Julia set of fλ$f_\lambda$ varies analytically over Ω$\Omega$ .