Search Results

As mentioned in my comment, $\psi_a(\sigma)$ clearly does not depend continuously on $a$; consider $f(s,a) := a\cdot s^2$ at $a=0$. For the question in your comment, note that this cannot remotely be …

The answer is "yes". (I assume that your curve is meant to be Jordan, i.e. a simple closed curve? Otherwise, some more thought would need to be put into the argument.) If $\gamma$ is the unit circle, …

Let me expand on the answer by Alex a little bit, and give a specific example. The question of whether such rational maps exist, and if yes, how many, goes back to Hurwitz ("Über Riemann’sche Fläche …

Your map is automatically going to be expanding at every point with respect to the hyperbolic metric (assuming $\Omega'\subsetneq\Omega$), and uniformly expanding if $\Omega'$ is compactly contained i …

I like to advocate differentiating between the singularities of the inverse function, which are denoted by $\operatorname{sing}(f^{-1})$, and the set of singular values of $f$, which I denote by $S(f) …

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 …

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 ( …

The paper of Detlef Bargmann, "Normal families of covering maps" in Journal d'Analyse, 2001, is relevant to this question; see http://dx.doi.org/10.1007/BF02788084 . His Theorem 1 implies, in particul …

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 …

(EDIT 1. Images have been added below.) As far as I understand, any non-rectifiable curve is not chord arc. So any such curve will do, e.g. the Julia set of $z^2+c$ for $c\neq 0$ in the main cardio …

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 …

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 …