# Questions tagged [complex-dynamics]

Dynamics of holomorphic transformations; Mandelbrot and Julia sets.

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This question is related to a classification of rational maps in terms of properties of their Julia set. Let $f= z^2 + c$, with $c\in \mathbb{C}$ be a quadratic polynomial such that its Julia set $J(... 0 votes 1 answer 113 views ### Rigid motions between two spheres [closed] It has been well known that every Mobius transformation can be constructed by stereographi projection of the complex plane onto a sphere, followed by a rigid motion of the sphere and projection back ... 3 votes 0 answers 74 views ### Does there exist a Runge Fatou-Bieberbach in each Fatou-Bieberbach domain? A Fatou-Bieberbach domain$\Omega \subseteq \mathbb{C}^n$is a domain that is a proper subset of$\mathbb{C}^n$and is biholomorphic to$\mathbb{C}^n$. A domain is said to be Runge if for each ... 1 vote 1 answer 146 views ### A singular holomorphic foliation of$\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$. 8 votes 2 answers 872 views ### Dynamics of Riemann zeta function Has the dynamics of the Riemann zeta function been studied? By dynamics I mean the limiting behavior of the sequence of iterates$s, \zeta (s), \zeta (\zeta (s)), \zeta (\zeta (\zeta (s)))\dots $for ... 3 votes 1 answer 140 views ### 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$n=k=3$, ... 1 vote 0 answers 51 views ### 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 ... 1 vote 1 answer 419 views ### 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 ... 0 votes 1 answer 245 views ### 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 ... 0 votes 1 answer 73 views ### 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$z_{n+1} = z_nz_n+c$... 3 votes 1 answer 111 views ### Decay of the binomial expansion of$f^{\circ k}$Suppose$f$is a holomorphic function in a neighborhood of zero fixing zero. Suppose$f'(0) = \lambda$and$0<\lambda < 1$. It's not so hard to prove that$f^{\circ k}(z) = f(f(\ldots\text{$k$ ...
It is a commonly-referenced result about certain rational maps acting on $\mathbb{\hat{C}}$ that they are expanding on a neighborhood of their Julia sets. A sufficient condition to be expanding is ...