I am reading the research article *"The Hausdorff dimension of the boundary of Mandelbrot set and Julia sets"* by Shishikura. The following two definitions are given without any examples in this paper. Therefore to understand the following definitions, I need examples of these two definitions:


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


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


Let $\Lambda$ be open set of $\mathbb{C}$. A family $\{f\_\lambda : \lambda \in \Lambda \}$ of rational maps is $J$-Stable at $\lambda\_0 \in \Lambda$, if there exists a continuous map $h : \Lambda ^{'} \times J(f\_{\lambda\_{0}}) \rightarrow \mathbb{C}\_{\infty}$, such that $\Lambda ^{'}$ is neighborhood of $\lambda\_0$ in $\Lambda$, $h\_{\lambda} \equiv h(\lambda,.)$ is conjugacy from $(J(f\_{\lambda\_{0}}),f\_{\lambda\_{0}})$ to $(J(f\_{\lambda}), f\_{\lambda})$ and $h\_{\lambda\_{0}} = \mathrm{id}\_{J(f\_{\lambda\_{0}})}$.


Where $J(f)$ denotes the Julia set of $f$ and in both definitions $\mathbb{C}\_{\infty}$ denotes the Riemann sphere.


To get the example of second definition I tried to work with the quadratic family $P(z) = z^2 + c$. But didn't get anything.


Note that I am not aware about hyperbolic dynamics. After studying some general theory of complex dynamics and some great examples in it. I am studying this research article to understand the proof that Hausdorff dimension of the boundary of the Mandelbrot set is $2$.



Examples of hyperbolic set and J-stable sets
Hyperbolic functions - for example, quadratic polynomials with an attractive periodic point - are examples of maps that are J-stable. The notion of J-stability arises from the famous article of Mañe, Sad and Sullivan. It is discussed in McMullen's book on renormalisation.


The trivial examples of hyperbolic sets are repelling periodic orbits. You can think of hyperbolic sets as a generalisation of this to more general repelling subsets of the Julia set. The Julia set of a hyperbolic map is also a hyperbolic set. More generally, any compact forward-invariant set disjoint from the closure of the critical orbits is a hyperbolic set.


There is a connection between the two notions : A hyperbolic set will move holomorphically in a neighbourhood in parameter space. This is why they are relevant to Shishikuras proof: The main point is that there are parameters for which there are hyperbolic sets of dimension arbitrarily close to 2. Then you find corresponding sets of parameters via the corresponding holomorphic motions.


