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As Adam notes, having two different attracting orbits is an open property. (Moreover, these maps are stable, meaning that nearby functions have conjugate dynamics on their Julia sets.) He also gives a …

The answer to both of your questions is negative, even if you replace $q^{-n}$ by any other sequence $(a_n)$ of natural numbers. The reason is, broadly speaking, that you may have of course have map …

It is an open problem whether this set has Lebesgue measure 0. Indeed, the question whether the boundary of the Mandelbrot set has measure zero remains open, and the set of finitely renormalisable par …

Your question is rather long and a little bit difficult to parse (for example, in the fifth paragraph, you appear to mean 'critical' rather than 'fixed' points), so I apologize for not having all of i …

In my opinion, the question is completely arbitrary. There is no reason to expect relationships between the dynamics of $f$ and that of its derivative. Their relationship will even change under affine …

Clearly if $F(R)$ is connected, then $F(R)$ consists of a single attracting or parabolic basin. Both cases can occur. Indeed, if you consider the slice $\operatorname{Per}_1(1)$ of quadratic rationa …

I think for multiple roots this is clear. At a multiple root $z_0$, you have an attracting fixed point. Take a small disc $D$ around this root; then (as you note) for a suitable perturbation $\tilde{p …

The question of when the Julia set of a function depends continuously on the parameters is well-studied. For the case of polynomials, see Douady, Does a Julia set depend continuously on the Polynomial …

There is quite a lot of literature these days on the measurable dynamics of transcendental entire functions. You may be interested, in particular, in the works of Mariusz Urbanski and his co-authors ( …

It is a little bit difficult to answer the question as posed, because there is a question as to what you mean by "depth". One of the previous answers mentions Misiurewicz points - parameters where th …

For a polynomial, this is equivalent to asking whether there can be infinitely many external rays landing at a single point. This could happen only if the function has a Cremer point (i.e., a non-line …

These results (which are indeed cute - I hadn't seen them before) are well-explained by the theory of parabolic explosion, which is by now classical. Indeed, for the Mandelbrot set, I think that the r …

If you read French, the article "Sur la théorie d'Ahlfors des surfaces" by Duval may be interesting: http://arxiv.org/abs/1311.1589 . It covers the full Ahlfors theory, not just the islands theorem. …

The best way to study these questions is in the framework of computable analysis. I will take your rational function as being given by oracles for its coefficients and for its critical points. (That i …

To put the existing answer in context, you will never get summability if your function is holomorphic at $0$ and $f'(0)=1$. In particular, the example given in your question is actually also not summa …