Consider a holomorphic family (f_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.
Bianchi et al. (Sun,) studied this question.