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arXiv · 2409.01361

Conformal measures of (anti)holomorphic correspondences

Nils Hemmingsson·Xiaoran Li·Zhiqiang Li
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Abstract

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 $1\leq \delta_{\operatorname{crit}}(x) <+\infty,$ the correspondence $F$ is (relatively) hyperbolic on the limit set $\Lambda_+(x)$, and $\Lambda_+(x)$ is minimal, then $\Lambda_+(x)$ admits a non-atomic conformal measure for $F$ and the Hausdorff dimension of $\Lambda_+(x)$ 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 $f$.

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Nils Hemmingsson, Xiaoran Li, Zhiqiang Li. 2024-09-02. Conformal measures of (anti)holomorphic correspondences. https://arxiv.org/abs/2409.01361

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