# Questions tagged [complex-dynamics]

Dynamics of holomorphic transformations; Mandelbrot and Julia sets.

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### Critical points of polarized endomorphisms of algebraic varieties

My main question is the following: Let $f: \mathbb{CP}^n \to \mathbb{CP}^n$ be a holomorphic endomorphism of degree $d \ge 2$ of $\mathbb{CP}^n$ . 1. Let $X \subset \mathbb{CP}^n$ be an ...
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### Rigid motions between two spheres [closed]

It has been well known that every Mobius transformation can be constructed by stereographi projection of the complex plane onto a sphere, followed by a rigid motion of the sphere and projection back ...
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### Does there exist a Runge Fatou-Bieberbach in each Fatou-Bieberbach domain?

A Fatou-Bieberbach domain $\Omega \subseteq \mathbb{C}^n$ is a domain that is a proper subset of $\mathbb{C}^n$ and is biholomorphic to $\mathbb{C}^n$. A domain is said to be Runge if for each ...
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### A singular holomorphic foliation of $\mathbb{C}^2$ with a bounded leaf

Is there a polynomial vector field on $\mathbb{C}^2$ which possesses a bounded regular leaf? By bounded, I mean a bounded subset of $\mathbb{C}^2$.
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### Dynamics of Riemann zeta function

Has the dynamics of the Riemann zeta function been studied? By dynamics I mean the limiting behavior of the sequence of iterates $s, \zeta (s), \zeta (\zeta (s)), \zeta (\zeta (\zeta (s)))\dots$ for ...
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### Symmetries for Julia sets of perturbations of polynomial maps

This is a naive question. Consider the Julia sets of the map $$z \mapsto z^n + \lambda / z^k$$ with $z,\lambda \in \mathbb{C}$, and the exponents $n,k \in \mathbb{N}$. For example, for $n=k=3$, ...
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### Uniform convergence of holomorphic automorphisms

Let $X$ be a complete Kobayashi hyperbolic complex manifold. It is well-known that the automorphism group of $X$ is a real Lie group where the topology on the automorphim group is the compact-open ...
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### Formal group law and Koenigs function conjecture?

Let $f(x,y)$ be a symmetric real function and a formal group law $$G(x + y) = f(G(x),G(y)). \tag{1}$$ Consider the equation $$h(2x) = f(h(x),h(x)) = A(h(x)). \tag{2}$$ This equation has many ...
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### About strange invariant set of the Lozi mappings

Consider the Tent map: $f_{μ}(x)=μx$, if $x<0.5$ and $f_{μ}(x)=μ(1-x)$ if $x≥0.5$. In this page (https://en.wikipedia.org/wiki/Tent_map) it was stated that: If $μ$ is greater than $2$ the map's ...
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### On the 2002 paper “Dynamics of polynomial automorphisms of $\mathbb{C}^k$” by Guedj and Sibony

I desperately need to read the paper  before meeting a would-be supervisor, but with limited undergraduate knowledge that I have like Aluffi's Algebra and Churchill's Complex Analysis, not even one ...
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### non linear operator over the mandelbrot set [closed]

I write here because Google Scholar does not give me feedbacks. The Mandelbrot set M could be defined as the set of all the complex plane point c where the recurrent sequences $z_{n+1} = z_nz_n+c$ ...
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### Newton method and Siegel disks

I am looking for a degree 3 polynomial $P$ whose associated Newton's method $z \mapsto z - P(z)/P'(z)$ has a Siegel disk. Is there an explicit example of such polynomial $P$?
For a polynomial $P$ of degree $n$ with real coefficients and with $n$ distinct real roots, the Newton's method $z_{n+1} = z_n - {P(z_n) \over P'(z_n)}$ converges for almost all initial values $z_0$ ...