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Post-critically pre-periodic quadratic polynomials, i.e. those for which the orbit of the critical point $0$ is pre-periodic, are well-known to be dense in the boundary of the Mandelbrot set. (This is …

To add a little bit more background in addition to Robert's answer: The first thing to clarify is what you mean by "diverges". Like Robert, I think you mean "does not converge to a finite value". How …

Let $K$ be the boundary of your circle domain $\Omega$. Let us suppose that every point of $K$ is accumulated on by a sequence of (pairwise different) circle components. Such an example is easy to c …

Removability with respect to homeomorphisms is different from the removability with respect to bounded functions mentioned in another answer. In particular, it is not necessary to have Hausdorff dime …

Let $L$ be the boundary of the strip $$ G := \{a + ib\colon a>0, -\pi < b < \pi\}, $$ parameterised in clockwise direction. Define $$ F_0\colon \mathbb{C}\setminus \overline{G}\to\mathbb{C}; \quad z\m …

To put the questions and answers in a wider context, the set of singular values $S(f)$ of an entire function $f$ is the set of points $w$ near which $f$ is not a covering map. That is, $w$ is regular …

If by "elementary method", you mean a proof that avoids using techniques from quasiconformal methods, then the short answer is "no". Giving a different proof of the No Wandering Domains (NWD) theorem, …

I hope that the following result by Bishop and myself might be "interesting": Every open Riemann surface is a cover over the sphere, branched over only three points. Equivalently, every open Riemann s …

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 …

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. …

As Adam stated, the Riemann mapping theorem provides the function that you are looking for. However, the Riemann mapping will not be given by an explicit formula, except in particularly simple cases. …

If $G\neq\mathbb{C}$ is a s.c. domain, then a conformal map to a disc extends analytically beyond the boundary if and only if $G$ is bounded by an analytic Jordan curve. In particular, if you were to …

To elaborate on my comment, choose three pairwise disjoint Jofrdan $U_1$, $U_2$, $U_3$ whose boundary passes through infinity, and let $\gamma_i\subset U_i$ be a curve to infinity for each $i$. Fix $\ …

I believe that your suggestion is correct: $\newcommand{\C}{\mathbb{C}}$ THEOREM. Suppose that $U,V\subset\mathbb{C}$ are domains, and that $f:U\to V$ is a conformal isomorphism. Then $f'$ has a ( …

Further to Ben's answer, it might be useful to picture the situation in $\mathbb{C}$. (Of course in $\mathbb{C}$ every domain is a domain of holomorphy, but we can still exhibit the same phenomenon th …