I am studying one of the research article "The Hausdorff Dimension of the Boundary of the Mandelbrot Set and Julia Sets" by Mitsuhiro Shishikura. There is one section where he gave the definition of Hyperbolic set of $\mathbb{C}_{\infty}$, where $\mathbb{C}_{\infty}$ is Riemann Sphere, as following
Let $f$ be rational map on $\mathbb{C}_{\infty}$. A closed subset $X$ of $\mathbb{C}_{\infty}$ is called Hyperbolic subset for $f$ if (1) $f(X) \subset X$ and (2) If there exist positive constant $c$ and $\kappa$ $> 1$ such that $\lVert (f^n)^{'} \rVert \geq c \kappa^{n}$ on $X$ for $n \geq 0$.Where $\rVert . \rVert$ denotes the norm of derivative with respect to the spherical metric on $\mathbb{C}_{\infty}$.
Here in this definition, I am not able to figure out how he defined the norm of function (here norm of derivative of $f$) with respect to the spherical metric on $\mathbb{C}_{\infty}$.
I am thinking that he defined the sup norm on the set of all function from the $\mathbb{C}_{\infty}$ to $\mathbb{C}_{\infty}$ (by some convention) as $\lVert f \rVert = sup$ $|f(x)|$ , where supremum is taken over $X$. To make the sense of use of spherical metric on $\mathbb{C}_{\infty}$, ( convention is that) replace $|f(x)|$ by $d(f(x) , 0)$, where d is spherical metric on $\mathbb{C}_{\infty}$.
It is just my guess I might be completely wrong. For the reference, I am attaching the image of that page.
No, this is not the meaning. You should think instead of $S^2$ with the spherical metric as a Riemannian manifold, with a norm defined on tangent spaces of $S^2$ at various points. For a smooth mapping $F: S^2\to S^2$, the derivative $F'$ is a map of the tangent bundle of $S^2$ to itself (not a function $S^2\to S^2$!). In particular, at each point $z\in S^2$ we have a linear map of normed vector spaces $$ F'(z): T_zS^2\to T_{F(z)}S^2. $$ (In this special case, the map is complex linear, but it does not really matter.) Then $||F'(z)||$ is just the operator norm of that linear map: $$ ||F'(z)||=\max \{||F'(z)(u)||_{F(z)}: u\in T_zS^2, ||u||_{z}=1\}. $$ Lastly, $$ ||F'||_X=\max_{z\in X} ||F'(z)|| $$ where $X$ is a subset of $S^2$. Thus, the expression ``$||F'||\ge C$ on $X$'' in the paper should be understood as $$ ||F'||_X\ge C. $$