Analyticity of the Hausdorff dimension and metric structures on Misiurewicz families of polynomials
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908492835586048 |
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| author | Bianchi, Fabrizio He, Yan Mary |
| author_facet | Bianchi, Fabrizio He, Yan Mary |
| contents | Consider a holomorphic family $(f_λ)_{λ\in Λ}$ of polynomial maps on $\mathbb C$ 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 $λ\in Ω$, 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 $\langle \cdot, \cdot \rangle_G$. 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 $Ω$. 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 $Ω$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_12493 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analyticity of the Hausdorff dimension and metric structures on Misiurewicz families of polynomials Bianchi, Fabrizio He, Yan Mary Dynamical Systems Complex Variables Consider a holomorphic family $(f_λ)_{λ\in Λ}$ of polynomial maps on $\mathbb C$ 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 $λ\in Ω$, 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 $\langle \cdot, \cdot \rangle_G$. 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 $Ω$. 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 $Ω$. |
| title | Analyticity of the Hausdorff dimension and metric structures on Misiurewicz families of polynomials |
| topic | Dynamical Systems Complex Variables |
| url | https://arxiv.org/abs/2508.12493 |