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

Consider a holomorphic family $(f_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }}$ 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 $\mathrm{\Omega }$ be a bounded stable component of $\mathrm{\Lambda }$ with the property that, for all $\lambda \in \mathrm{\Omega }$, all the other critical points of $f_{\lambda }$ belong to attracting basins. In this paper, we introduce a dynamically meaningful geometry on $\mathrm{\Omega }$ by constructing a natural path metric on $\mathrm{\Omega }$ coming from a 2-form $⟨\cdot ,\cdot {⟩}_G$. 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 $⟨\cdot ,\cdot {⟩}_G$ on the unit tangent bundle of $\mathrm{\Omega }$. As part of our construction, we recover a result of Skorulski and Urbański stating that the Hausdorff dimension of the Julia set of $f_{\lambda }$ varies analytically over $\mathrm{\Omega }$.