1
vote
0answers
77 views

Algebraic set given by sequence of polynomials

When working on some problem, I have end up with a following situation. Suppose $P(z)=z^d+a_{d-1}z^{d-1}+\ldots+a_1z+a_0$ is a complex polynomial $d\geq2$ and that $\gamma$ and $\delta$ are non-zero ...
1
vote
1answer
78 views

Normal family and arithmetic progression

It is basic fact that for a holomorphic (or mermorphic) map $f$, the family of iterates $\{f^n\}_{n=1}^{\infty}$, is normal if and only if $\{f^{mn}\}_{n=1}^{\infty}$, is normal $\forall m\geq 1$. ...
1
vote
0answers
41 views

A criterion or algorithm for polynomial which admits Markov partition on its Julia set

For a given polynomials $P(z)$, whether there exists general algorithm to check it admits a Markov partition on its Julia sets. (in finite computation time.) May be it is more difficult for the ...
2
votes
2answers
90 views

general variational principle for the Julia sets of mermorphic function?

I have seen some attempt in considering topological pressure for Julia sets of exponential function, and elliptic function. However, there exists few reference according to my knowledge? I want to ...
3
votes
1answer
163 views

Fixed points on Riemann surface

It is well known theorem that for a conformal mapping $\phi$ from a bounded and planar domain $\Omega$ to itself has three fixed points , then it must be identity mapping. However, I cannot find a ...
2
votes
2answers
145 views

Is the Hausdorff dimension $Dim_{H}(J(f))$ of the Julia set less than 2 for quadratic rational map?

Let $f(z)$ be a quadratic rational map with two Siegel disks which can be normalized to be $$f(z)=z\frac{z+e^{2\pi i\alpha}}{e^{2\pi i\beta}z+1}.$$ If one of the ratation numbers $\alpha$ and $\beta$ ...
4
votes
2answers
599 views

Is complex analytic extension of real-analytic diffeomorphism a diffeomorphism ?

Hi, my question is : Let $D$ be the open unit disk in $\mathbb{C}=R^2$ and $f:D\to D$ be a real-analytic diffeomorphism. Let us think of the canonical embedding : $\mathbb{C}=R^2\subset \mathbb{C^2}. ...