Semi-hyperbolic rational maps and size of Fatou components

Abstract

Recently, Merenkov and Sabitova introduced the notion of a homogeneous planar set. Using this notion they proved a result for Sierpiński carpet Julia sets of hyperbolic rational maps that relates the diameters of the peripheral circles to the Hausdorff dimension of the Julia set. We extend this theorem to Julia sets (not necessarily Sierpiński carpets) of semi-hyperbolic rational maps, and prove a stronger version of the theorem that was conjectured by Merenkov and Sabitova.

Combining rational maps and Kleinian groups via orbit equivalence

“…This implies that, given any $$0<\varepsilon \leqslant \varepsilon _0$$, $$$$\begin{align} \sigma {\left(\lbrace z\in \mathcal {U}^{\prime }\colon |\mu (z)|\geqslant 1-\varepsilon \rbrace \right)}\leqslant L^2(\operatorname{diam}{\mathcal {U}^{\prime }})^2\sigma {\left(\lbrace z\in P_\Gamma ^{\circ r}(\mathcal {U}^{\prime })\colon |\mu (z)|\geqslant 1-\varepsilon \rbrace \right)}. \end{align}$$$$Moreover, since all the Fatou components $$\mathcal {U}^{\prime }$$ are uniform John domains [38, Proposition 10], it follows from [50, p. 444] that there exists a constant $$C_1>0$$ such that $$$$\begin{align} (\operatorname{diam}{\mathcal {U}^{\prime }})^2\leqslant C_1\sigma {\left(\mathcal {U}^{\prime }\right)}, \end{align}$$$$for all strictly preperiodic Fatou components $$\mathcal {U}^{\prime }$$ of $$P_\Gamma$$. Putting inequalities (), (), (), and () together, we obtain our ...…”

On the Hausdorff dimension of the residual set of a packing by smooth curves

“…For $$M>0$$, we say that two bounded regions $$D_1,D_2$$ are $$M$$‐relatively separated if $$\begin{align*} \Delta (D_1,D_2) \coloneqq \frac{\operatorname{{\mathrm{dist}}}(D_1,D_2)}{\min \lbrace \operatorname{{\mathrm{diam}}}(D_1),\operatorname{{\mathrm{diam}}}(D_2)\rbrace } \geqslant M. \end{align*}$$Moreover, we say that a Jordan curve $$C$$ is an $$M$$‐quasicircle if for any two points $$z,w\in C$$ there exists an arc $$A$$ of $$C$$ connecting $$z$$ and $$w$$ such that $$\operatorname{{\mathrm{diam}}}(A)\leqslant M|z-w|$$. It is proved in [18, Proposition 4.1] that the residual set of a planar packing $$\mathcal {P}$$ by Jordan regions …”