I am reading the research article *"The Hausdorff dimension of the boundary of Mandelbrot set and Julia sets"* by Shishikura. In his article, he defined hyperbolic sets and hyperbolic dimensions for any rational map of $\overline{\mathbb{C}}$ onto itself, where $\overline{\mathbb{C}}$ is Riemann sphere.

Let $f$ be rational map on $\overline{\mathbb{C}}$. A closed subset $E$ of $\overline{\mathbb{C}}$ is called a hyperbolic subset for $f$ if

- $f(E) \subset E$ and
- there exist a positive constant $c$ and $\kappa$ $> 1$ such that $\lVert (f^n)' \rVert \geq c \kappa^{n}$ on $E$ for $n \geq 0$. Here $\rVert \cdot \rVert$ denotes the norm of derivative with respect to the spherical metric on $\overline{\mathbb{C}}$.

The hyperbolic dimension of $f$ is defined as \begin{align} \operatorname{hyp-dim}(f):= \sup\{\operatorname{H-dim}(E) : E \; \text{is hyperbolic set of $f$} \;\} \end{align} where $\operatorname{H-dim}(E)$ is Hausdorff dimension of $E$.

Since he didn't give any example of this. Therefore I want to work with some examples for these definitions. Suppose I want to find the hyperbolic dimension of one of the simplest nonlinear rational map $f(z) = z^2$. How to proceed? I believe that then I need to tackle all the hyperbolic sets for $f(z) = z^2$. But this does not seem trivial. I am looking forward for the help regarding finding the hyperbolic dimension of $f(z) = z^2$.

`'`

in TeX already has a superscript built in (it essentially translates to`^{\prime}`

, so you almost never want`^{'}`

. Compare $f' f^\prime f^{'}$`f' f^\prime f^{'}`

. I edited accordingly. $\endgroup$