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

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