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 dimension for any rational map of $\overline{\mathbb{C}}$ onto itself.

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

1. $f(E) \subset E$ and
2. 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 $f(z) = z^2$.

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

1. $f(E) \subset E$ and
2. 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 $f(z) = z^2$.

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 dimension for any rational map of $\overline{\mathbb{C}}$ onto itself.

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

1. $f(E) \subset E$ and
2. 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 $\rVert . \rVert$ 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} hyp-dim(f):= \sup\{H-dim(E) : E \; \text{is hyperbolic set of f} \;\} \end{align} Where $H-dim(E)$ 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 $f(z) = z^2$. Thanks in advance.

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 dimension for any rational map of $\overline{\mathbb{C}}$ onto itself.

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

1. $f(E) \subset E$ and
2. 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 $f(z) = z^2$.