I read books The Beauty of Fractals --- Images of complex dynamical systems, in one chapter it shows properties of Julia Set as follows (we just consider rational functions on Riemann Surface where Julia set is denoted $J_R$),

  • $ J_R \neq \emptyset$ and contains more than countably many points.
  • The Julia sets are completely invariant, which means $R(J_R)=R^{-1}(J_R)=J_R$.
  • The Julia sets of $R$ and $R^k$, $k= 1,2, \cdots ,$ are identical, where $R^k$ is $k$ times iterations of rational function $R$.
  • For any $x\in J_R$, the inverse orbit $Or^-(x)$ is dense in $J_R$, where $Or^-(x_0)=\{x\in\overline {\mathbb C}:R^k(x)=x_0, k=0,1,2\cdots\}$.

Now I have a proof from some paper of how to prove $J_R$ is not an empty set in the point of topological degree, I cannot understand it. I want some more accessible ways to prove it, better not using topological degree. I have no idea how to start to show that $J_R$ contains more than countably many points, should I start from proving it is infinite and not countable?

$J_R$ is left invariant about the function $R$, it is easy to understand, but how about $R^{-1}(J_R)=J_R$?

Third property seems easy to understand for me.

Last one, I do not know how to start, either, should I use definition of $Dense$?

Any hint will be appreciated.


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