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Conformal measures of (anti)holomorphic correspondences

Nils Hemmingsson, Xiaoran Li, Zhiqiang Li

TL;DR

This work develops a Patterson–Sullivan framework for conformal measures on limit sets of (anti)holomorphic correspondences, extending classical constructions for rational maps and Kleinian groups. By defining forward limit sets $Λ_+(x)$, the critical exponent $δ_{\operatorname{crit}}(x)$ via Poincaré series, and notions of relative hyperbolicity and minimality, the authors prove the existence of non‑atomic $δ$‑conformal measures when $1 \le δ_{\operatorname{crit}}(x) < ∞$ and the dynamics on $Λ_+(x)$ is relatively hyperbolic and minimal, with $\,\operatorname{HD}(Λ_+(x))<2$. They apply these results to LLMM and Bullett–Penrose correspondences, showing that in natural parameter regimes these limit sets support non‑atomic conformal measures with $1 \le δ < 2$ and Hausdorff dimension bounded by $δ$, providing a deep link between geometric, dynamical, and fractal properties of correspondences. The work also develops tools to handle critical points within holomorphic correspondences and establishes that, in the relatively hyperbolic/minimal setting, open conformal measures exhibit robust structure and that the limit sets have zero area, advancing understanding of fractal geometry in this broader dynamical context.

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 $1\leq δ_{\operatorname{crit}}(x) <+\infty,$ 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 $f$.

Conformal measures of (anti)holomorphic correspondences

TL;DR

This work develops a Patterson–Sullivan framework for conformal measures on limit sets of (anti)holomorphic correspondences, extending classical constructions for rational maps and Kleinian groups. By defining forward limit sets , the critical exponent via Poincaré series, and notions of relative hyperbolicity and minimality, the authors prove the existence of non‑atomic ‑conformal measures when and the dynamics on is relatively hyperbolic and minimal, with . They apply these results to LLMM and Bullett–Penrose correspondences, showing that in natural parameter regimes these limit sets support non‑atomic conformal measures with and Hausdorff dimension bounded by , providing a deep link between geometric, dynamical, and fractal properties of correspondences. The work also develops tools to handle critical points within holomorphic correspondences and establishes that, in the relatively hyperbolic/minimal setting, open conformal measures exhibit robust structure and that the limit sets have zero area, advancing understanding of fractal geometry in this broader dynamical context.

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 the correspondence is (relatively) hyperbolic on the limit set , and is minimal, then admits a non-atomic conformal measure for and the Hausdorff dimension of is strictly less than 2. As a special case, this shows that for a parameter in the interior of a hyperbolic component of the modular Mandelbrot set, the limit set of the Bullett--Penrose correspondence 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 .
Paper Structure (8 sections, 29 theorems, 85 equations, 2 figures)

This paper contains 8 sections, 29 theorems, 85 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Singular points, minimal forward limit sets, and relative hyperbolicity
  3. Examples of minimal forward limit sets on which the correspondence is relatively hyperbolic
  4. Forward limit sets of LLMM correspondences
  5. Limit sets of Bullett--Penrose correspondences
  6. Constructing conformal measures
  7. Conformal measures for relatively hyperbolic correspondences on minimal forward limit sets
  8. Sufficient conditions for critical exponent greater than 1

Key Result

Theorem A

Let $f\in \mathcal{M}$ and $F$ be the LLMM correspondence defined by $f$. Suppose that the map $R$, appearing in Definition def:finM, has an attracting periodic orbit in $\widehat{\mathbb{C}}\smallsetminus \mathcal{B}(R)$. Then there exists a non-atomic $\delta$-conformal measure for $F$ and $\Lambd

Figures (2)

  • Figure 2.1: Zoomed-in picture of $\Lambda_+(x)$ near the parabolic fixed point $\omega=1/2$, indicated by a blue dot, for $F(z)\coloneqq R^{-1}(z)$ where $R\coloneqq z^2+1/4$. As long as $x\neq \infty$, $\Lambda_+(x)$ equals the Julia set of $R$, which is the common boundary of the black and the colored regions. The set $U$ in Definition \ref{['def:Omega-attracting']} can be taken as the interior of the red circle sector, provided that it is chosen sufficiently small.
  • Figure 3.1: The dynamics of $F_1$ on a neighborhood of the parabolic fixed point $z_0 =1$. The purple curve is a part of the unit circle. $\mathbf{v}_1,\, \mathbf{v}_3,\, \dots ,\, \mathbf{v}_{2n-1}$ are all attracting directions, and $\mathbf{v}_2,\, \mathbf{v}_4,\, \dots ,\, \mathbf{v}_{2n}$ are all repelling directions.

Theorems & Definitions (78)

  • Definition 1.1
  • Definition 1.2
  • Theorem A
  • Theorem B
  • Definition 1.3
  • Remark 1.4
  • Theorem C
  • Remark 1.5
  • Theorem D
  • Definition 2.1
  • ...and 68 more