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

Consider a holomorphic family (f_λ)λ∈ Λ of polynomial maps on C with the property that a critical point of f_λ is persistently preperiodic to a repelling periodic point of f_λ. Let $Ω$ be a bounded stable component of $Λ$ with the property that, for all λ∈ Ω, all the other critical points of f_λ belong to attracting basins. In this paper, we introduce a dynamically meaningful geometry on $Ω$ by constructing a natural path metric on $Ω$ coming from a 2-form ·, · 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 on the unit tangent bundle of $Ω$. As part of our construction, we recover a result of Skorulski and Urbański stating that the Hausdorff dimension of the Julia set of f_λ varies analytically over $Ω$.