# Tagged Questions

**5**

votes

**3**answers

114 views

### Is there a effective computational criterion to all periodic points of a rational function are repelling.

I came up with a question to know the fatou component of of some types of rational function. In some sense, I may need to give a computational criterion to existence of attracting periodic basin for a ...

**2**

votes

**2**answers

97 views

### Original article about a theorem of Cartan on iterations of analytic functions

I'd like to know in which paper of H. Cartan I could find the following theorem :
Let $\Omega$ be a connected, open and bounded subset of $\mathbb{C}$. Let $a \in \Omega$ and $f \in ...

**1**

vote

**1**answer

65 views

### A question for the inverse orbit in the construction of conformal measure

Recently, I read a theorem of existence of conformal measure for the rational map.
I did not understand two places in the proof. The author claims that
there exists an open set $V\subset ...

**4**

votes

**1**answer

171 views

### Contractibility of connected holomorphic dynamics?

Let $f$ be a function, holomorphic in $\mathbb{C}$, and $K(f)$ its non-escaping set :
$$K(f) = \{ z \in \mathbb{C} : f^{(k)}(z) \nrightarrow_{k \to \infty} \infty \} $$
Question : If $K(f)$ ...

**3**

votes

**0**answers

238 views

### A Dedekind Eta trajectory / horocyclic flow (Reference request)

I've been exploring the composition of essentially the Dedekind $\eta$-function with
parabolic Möbius transformations,
...

**6**

votes

**0**answers

190 views

### Invariant curves of rational functions

Let $\gamma$ be a Jordan analytic curve on the Riemann sphere, and $f$ a rational function
of degree at least 2 which
maps $\gamma$ onto itself homeomorphically. The following examples of such ...

**23**

votes

**3**answers

793 views

### Rational functions with a common iterate

Let $f$ and $g$ be two rational functions. To avoid trivialities, we suppose that their degrees are
at least $2$. We say that they have a common iterate if $f^m=g^n$ for some positive integers $m,n$,
...

**4**

votes

**1**answer

384 views

### Beautiful examples of arc-like continua

A continuum is a nonempty compact, connected metric space.
A continuum $X$ is called arc-like if, for every $\varepsilon>0$, there is a continuous and surjective function $f:X\to [0,1]$ such that ...

**5**

votes

**1**answer

310 views

### Mandelbrot set and analytic functions such that $f(az)=f(z)^2+c$

It is well known that the function $f(z)=2\cos(\sqrt {-z})$ (or more accurately the entire function $f(z)=2\sum_{n=0}^\infty \frac{z^n}{(2n)!}$) satisfies such a functional equation, i.e. $f(4z)= ...

**3**

votes

**2**answers

297 views

### maximum of two plurisubharmonic

Let $u,v$ be two plurisubharmonic functions in a domain $\Omega\subset \mathbb{C}^n.$ Then
$w=\max$ {$u,v$} is plurisubharmonic. The support of $dd^c w$ is unclear in {$z: u(z)=v(z)$}
How to ...

**2**

votes

**1**answer

550 views

### How to calculate Dr. Curt McMullen's expanding eigenvalues for totally degenerate groups?

What is required in order to derive the expanding eigenvalues of Dr. Curt McMullen's torus orbifold bundles over the circle and the corresponding totally degenerate groups, as presented in Section 3.7 ...

**3**

votes

**2**answers

458 views

### self-similarity of a dendrite fractal

The Julia set of the map $z \mapsto z^2+i$ is a dendrite fractal. I would like to know which affine maps (other than identity) map this region to a subset of itself. I imagine there are two three ...

**4**

votes

**2**answers

247 views

### Non-trivial surjective holomorphic selfmap of compact Kähler manifold of $\operatorname{dim} \geq 3$?

I am finding some nontrivial examples of surjective holomorphic selfmap of compact Kähler manifold of $\operatorname{dim} \geq 3$. That is, find $X$ a compact Kähler manifold of $\operatorname{dim} X ...

**11**

votes

**4**answers

1k views

### Routh-Hurwitz for eigenvalues

The Routh-Hurwitz criterion provides a convenient test, even for hand calculation, of whether a polynomial with real coefficients has all its roots in the left half plane. I'm wondering about a ...

**8**

votes

**0**answers

362 views

### Parametrisations for Null Temperature Functions: nonuniqueness of solutions to the Heat Equation

Disclaimer I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks!
Definition ...

**4**

votes

**3**answers

713 views

### How can one express the Dedekind eta function as a sum over the lattice?

The Dedekind eta function $\eta(\tau)$ can be regarded as a formula which assigns a number to a lattice $L \subset \mathbb{C}$. The algorithm is: rotate the lattice so that one of its basis vectors ...

**94**

votes

**1**answer

9k views

### What are the shapes of rational functions?

I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...

**2**

votes

**1**answer

357 views

### Infinite-dimensional complex polynomial or rational Lie algebras and their pseudogroups

In studying the transformation groups generated by holomorphic vector fields V(z) d/dz on ℂ, I've noticed the (surely well-known) fact that the complex quadratic vector fields:
...