Dynamics of holomorphic transformations; Mandelbrot and Julia sets.

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7
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358 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 ...
6
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139 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 ...
3
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60 views

Is there an efficient way to visualize the bifurcation locus of this family of functions?

I have been trying to help out with this question from math.stackexchange. It concerns the family of functions: $$f_\alpha(z,w) = \frac{\alpha + z}{1 + w}.$$ and an iteration scheme: $$z_{n+1} = ...
3
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71 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 ...
3
votes
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343 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 ...
3
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229 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 ...
2
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91 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. ...
1
vote
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93 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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47 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 ...
0
votes
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77 views

Periodicities of a Complex Dynamical System

Consider A function $f:\mathbf{C}^2\rightarrow \mathbf{C}$ defined as $$f_{\alpha, \beta}(z,w)=\frac{\alpha}{z}+\frac{\beta}{w}$$ where $\alpha$ and $\beta$ both are complex number. It is easy to ...
0
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156 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 : ...