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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.01361 |
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| _version_ | 1866913658010861568 |
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| author | Hemmingsson, Nils Li, Xiaoran Li, Zhiqiang |
| author_facet | Hemmingsson, Nils Li, Xiaoran Li, Zhiqiang |
| contents | In this paper, we study the existence and properties of conformal measures on limit sets of (anti)holomorphic correspondences. We show that if the critical exponent satisfies $1\leq δ_{\operatorname{crit}}(x) <+\infty,$ the correspondence $F$ is (relatively) hyperbolic on the limit set $Λ_+(x)$, and $Λ_+(x)$ is minimal, then $Λ_+(x)$ admits a non-atomic conformal measure for $F$ and the Hausdorff dimension of $Λ_+(x)$ is strictly less than 2. As a special case, this shows that for a parameter $a$ in the interior of a hyperbolic component of the modular Mandelbrot set, the limit set of the Bullett--Penrose correspondence $F_a$ has a non-atomic conformal measure and its Hausdorff dimension is strictly less than 2. The same results hold for the LLMM correspondences, under some extra assumptions on its defining function $f$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01361 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conformal measures of (anti)holomorphic correspondences Hemmingsson, Nils Li, Xiaoran Li, Zhiqiang Dynamical Systems In this paper, we study the existence and properties of conformal measures on limit sets of (anti)holomorphic correspondences. We show that if the critical exponent satisfies $1\leq δ_{\operatorname{crit}}(x) <+\infty,$ the correspondence $F$ is (relatively) hyperbolic on the limit set $Λ_+(x)$, and $Λ_+(x)$ is minimal, then $Λ_+(x)$ admits a non-atomic conformal measure for $F$ and the Hausdorff dimension of $Λ_+(x)$ is strictly less than 2. As a special case, this shows that for a parameter $a$ in the interior of a hyperbolic component of the modular Mandelbrot set, the limit set of the Bullett--Penrose correspondence $F_a$ has a non-atomic conformal measure and its Hausdorff dimension is strictly less than 2. The same results hold for the LLMM correspondences, under some extra assumptions on its defining function $f$. |
| title | Conformal measures of (anti)holomorphic correspondences |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2409.01361 |