标题： 解析性 Hausdorff 维数和多项式 Misiurewicz 家族上的度量结构 显示英文标题

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

摘要： 考虑一个在$\mathbb C$上的多项式映射的全纯族$(f_\lambda)_{\lambda \in \Lambda}$，其性质是$f_\lambda$的一个临界点持续地预周期地指向$f_\lambda$的一个排斥周期点。 Let $\Omega$ be a bounded stable component of $\Lambda$ with the property that, for all $\lambda \in \Omega$, all the other critical points of $f_\lambda$ belong to attracting basins. In this paper, we introduce a dynamically meaningful geometry on $\Omega$ by constructing a natural path metric on $\Omega$ coming from a 2-form $\langle \cdot, \cdot \rangle_G$. Our construction uses thermodynamic formalism. 一个关键的要素是适应性转移算子在适当巴拿赫空间上的谱间隙，这也意味着$\langle \cdot, \cdot \rangle_G$在$\Omega$的单位切丛上的解析性。作为我们构造的一部分，我们恢复了 Skorulski 和 Urbański 的结果，该结果指出$f_\lambda$的 Julia 集的豪斯多夫维数在$\Omega$上解析变化。 显示英文摘要

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