A new perspective on the Sullivan dictionary via Assouad type dimensions and spectra

Abstract

The Sullivan dictionary provides a beautiful correspondence between Kleinian groups acting on hyperbolic space and rational maps of the extended complex plane. An especially direct correspondence exists concerning the dimension theory of the associated limit sets and Julia sets. In recent work we established formulae for the Assouad type dimensions and spectra for these fractal sets and certain conformal measures they support. This allows a rather more nuanced comparison of the two families in the context of dimension. In this expository article we discuss how these results provide new entries in the Sullivan dictionary, as well as revealing striking differences between the two settings.

Figures (4)

• Figure 1: Left: an example of a Kleinian limit set. Here $d=2$ and the boundary $\mathbb{S}^2$ has been identified with $\mathbb{R}^2 \cup \{\infty\}$. Parabolic points with rank 1 are easily identified. Right: an example of a parabolic Julia set. Parabolic points with petal number 4 are easily spotted. See the following section for definitions and notation.
• Figure 2: Left: a Kleinian limit set with $\delta=0.6$ and $k_{\min}=k_{\max}=1$. Right: a Julia set with $h=0.7$ and $p_{\max}=2$.
• Figure 3: Left: a Kleinian limit set with $\delta=1.9$ and $k_{\min}=k_{\max}=1$. Right: a Julia set with $h=1.4$ and $p_{\max}=4$.
• Figure 4: A Kleinian limit set with $\delta = 1.7$, $k_{\min}=1$ and $k_{\max}=2$. In the Julia setting we always have either ${\text{dim}}^{\theta}_\text{A} m = {\text{dim}}^{\theta}_\text{A} J(T)$ or ${\text{dim}}^{\theta}_\text{L} m = {\text{dim}}^{\theta}_\text{L} J(T)$, and so plots of this form are impossible in the Julia setting.