Conformal measures of (anti)holomorphic correspondences

TL;DR

This paper investigates conformal measures on limit sets of (anti)holomorphic correspondences, establishing conditions for their existence and properties, including Hausdorff dimension bounds, with applications to specific dynamical systems.

Contribution

It provides new criteria for the existence of conformal measures on limit sets of (anti)holomorphic correspondences and analyzes their Hausdorff dimensions under hyperbolicity conditions.

Findings

Conformal measures exist on limit sets when the critical exponent is finite and the correspondence is hyperbolic.

The Hausdorff dimension of these limit sets is strictly less than 2 under the given conditions.

Results apply to the Bullett--Penrose and LLMM correspondences within certain parameter regimes.

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 $f$.