Dynamics of holomorphic transformations; Mandelbrot and Julia sets.

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251 views

### degree of a rational map on infinitely connected fatou component

Given a rational map $f$ on the Riemann sphere, for their Fatou components, we can calculate the relations between the degree $k=\deg(f|_F)$, connectivity number $n=\mathrm{conn}(F)$ and number of ...

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345 views

### What is the “category of bifurcations”?

While reading the introduction to this paper by Curtis McMullen, I came to the following (bold added):
In this paper we show that every bifurcation set contains a copy of the boundary of the ...

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74 views

### When is a Newton basin fractal continuously determined by the roots of its polynomial?

Newton basin fractals are visualizations of the Julia sets of functions of the form:
$$f_p(z) = z - p(z)/p'(z)$$
where $p$ is a complex polynomial. My question is:
When is the Julia set, ...

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109 views

### Measure theoretic entropy

I don't know if this is an elementary question or not. In what follows all maps are continuous
Suppose that $P:\mathbb{C}\rightarrow\mathbb{C}$ is a complex polynomial of degree $d>1$ and let ...

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69 views

### Question about a length inequality in algebraic dynamics

Let $X$ be a Noetherian scheme. Let $f\colon X\rightarrow X$ be an integral self-morphism. If $x\in X$ is a closed point, I will write $\mathcal{F}_{1}^x$ for the coherent sheaf of ...

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332 views

### confusion about rational maps and Fatou components

Dear fellows,
I have come to another conclusion which must be wrong.
Let $f$ be a rational map and let $U$ be a connected but not simply connected open subset of the Fatou set such that $f(U)$ is ...

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219 views

### Picture of the set of discontinuity of degree 2 rational Julia sets

Let $Rat_d$ be the set of all rational fraction of degree $d$ and $X_d \subset Rat_d$ be the bifurcation locus of rational fractions of degree $d$, i.e. the closure of the set of discontinuity of the ...

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84 views

### Reachability in dynamic random graphs

There has been some advice on how to ask questions. So I reask my question.I have a problem related to graph theory, random graphs or random geometric graphs in special that really confuses me. ...

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87 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 ...

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45 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 ...

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145 views

### Limit sets of Fuchsian groups and relation between lifts to $H$ of homotopic maps between hyperbolic Riemann surfaces

Let $f,g : X \to Y$ be homotopic (quasiconformal) maps between hyperbolic Riemann surfaces $X,Y$. Consider their (unique) lifts $\tilde{f},\tilde{g}: H\to H$ , that fix $0,1,\infty $. My question is : ...