Questions tagged [complex-dynamics]

Dynamics of holomorphic transformations; Mandelbrot and Julia sets.

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4
votes
1answer
190 views

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 ...
3
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0answers
87 views

Exponential iterates of a complex number

Let $f:\mathbb C\to \mathbb C$ be defined by $f(z)=e^z-1$. Let $f^n$ denote the $n$-fold composition of $f$. In my new paper Erdős space in Julia sets I show that $$Z:=\{z\in \mathbb C:\lvert\...
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0answers
43 views

Locally connectedness and accessibility in $\mathbb{C}$

Suppose $\Omega$ to be a bounded area in the complex plane $\mathbb{C}$ with a locally connected boundary $\partial\Omega$, then every point of $\partial\Omega$ is accessible from Ω.Here accessibility ...
23
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2answers
685 views

Exponential towers of $i$'s

It's well known that the expression $i^i$ takes on an infinite set of values if we understand $w^z$ to mean any number of the form $\exp (z (\ln w + 2 \pi i n))$ where $\ln$ is a branch of the natural ...
10
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1answer
227 views

Convex Julia sets

Consider the classical Julia set $J_f$ associated with $f(z)=z^2+c$. Since $J_c$ is completely invariant, we know that $f^{-1}(J_f) \subseteq J_f$. Now, let $H_f$ be the convex hull of $J_f$. Is it ...
3
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3answers
211 views

Computing the maximum modulus

For each $a\in \mathbb C$ define $f_a:\mathbb C\to \mathbb C$ by $f_a(z)=\exp(z)+a$. I am primarily interested in real values $a\in (-\infty,-1)$. For each $r\in [0,\infty)$ define $M_a(r)=\max\{|f_a(...
5
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3answers
170 views

Fully invariant measures for rational functions

Let $f(z)$ be a rational function of degree $d \geq 2$, with complex coefficients. I am interested in fully invariant measures for the dynamical system $(\mathbb C_\infty,f)$, where $\mathbb C_\infty$ ...
5
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1answer
411 views

On complex dynamics in high dimensions

I am a fresh Ph.D student and I'm interested in complex dynamics in high dimensions. I have the following questions. What research directions are there in several complex dynamics and what problems ...
0
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1answer
105 views

Can the immediate basin of attraction of super-attracting fixed point at 0 of a polynomial contain non-zero roots?

Let $f$ be a polynomial with a super attracting fixed point at $x=0$. Can the immediate basin of attraction of the fixed point contain other roots? If so, please provide a specific example with the ...
1
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1answer
86 views

Is the set of non-escaping points in a Julia set always totally disconnected?

I am looking for examples of transcendental entire functions $f:\mathbb C\to \mathbb C$ such that the set of non-escaping points in the Julia set of $f$ is not totally disconnected. I denote this set $...
12
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2answers
360 views

Algorithm for computing external angles for the Mandelbrot set

Let $M$ be the Mandelbrot set: there exists a unique series $$ \psi(z) := z + \sum_{m=0}^{+\infty} b_m z^{-m} = z - \frac{1}{2} + \frac{1}{8} z^{-1} - \frac{1}{4} z^{-2} + \cdots $$ which defines a ...
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0answers
60 views

Extending the entire functions of hyperoperators

Addition, multiplication and exponentiation, the first three arithmetic operators, have very general domains and ranges. Also, iterated functions are often defined in Banach spaces. My question is ...
5
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0answers
194 views

Showing that a certain level set of a continuous family of holomorphic maps is locally path connected

I'm working with a continuous function $P: [0,1] \times W \to \mathbb{C}^n$, where $W \subset \mathbb{C}^n$ is an open, relatively compact ball centred at the origin. The map $P$ satisfies the ...
5
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0answers
90 views

wild julia sets

Using the Baire category theorem, we may show that most simple closed curves satisfy the following property: any segment between an interior point and an exterior point of the curve intersects the ...
1
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0answers
412 views

The mysterious numbers $ \frac{13}{20} $ and $20$?

Let $g(x) = x^6 - 30 x $ Let $h(x) = x^6 $ Let $f(x) = x^2 - 2 $ Let $r$ be a reduced fraction $0 < \frac{p}{q} < 2 $ with integers $p,q > 1$ Let $f_{n+1}(x) = f(f_n(x)) = f_n(f(x)) , ...
5
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2answers
505 views

Who proved that the Mandelbrot set's Julia sets are locally connected?

I'd be greatly interested in a reference to the respective article. Was it Douady? Julia? Hubbard? Fatou? Bonus question: Who gave the proof that can be found in the Orsay notes? EDIT: The question ...
13
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1answer
421 views

Is the set of escaping endpoints for $e^z-2$ completely metrizable?

Let $f:\mathbb C \to \mathbb C$ be the complex exponential $$f(z)=e^z-2.$$ It is known that $J(f)$, the Julia set of $f$, is a uncountable collection of disjoint rays (one-to-one continuous images ...
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0answers
46 views

Modulus estimate with intersecting annuli (quasi-additivity)

In general for annulus $A\subset \mathbb{C}$ if $A_{1},A_{2}....\subset A$ are disjoint annuli inside it, then we have $$mod(A)=\frac{1}{2\pi}\int_{A}\int_{A} \frac{1}{|z|^{2}}dz>\frac{1}{2\pi}\...
5
votes
1answer
234 views

An entire function all whose forward orbits are bounded

Edit: I revise the question according to the comment of Gabe Conant. What is an example of a non constant entire function $f:\mathbb{C}\to \mathbb{C}$ which satisfy the following?: For every $...
5
votes
2answers
350 views

Smooth Julia set for quadratic polynomials

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
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1answer
78 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
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0answers
63 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
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1answer
135 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 $\mathbb{C}^2$.
7
votes
2answers
563 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
1answer
131 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
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0answers
43 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 ...
2
votes
1answer
368 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 ...
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0answers
61 views

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 ...
1
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1answer
224 views

On the 2002 paper “Dynamics of polynomial automorphisms of $\mathbb{C}^k$” by Guedj and Sibony

I desperately need to read the paper [1] 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
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1answer
68 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
1answer
101 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$ ...
2
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0answers
78 views

How to compute expansion factors for hyperbolic rational maps?

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 ...
3
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0answers
74 views

Flow of zeros in the shifted exponential generating function?

Given a sequence $a_n$ (of real numbers, described more fully below), one may define the exponential generating function (on the complex plane) as $E(z)=\sum_{n=0}^\infty a_n z^n/n!$. The derivatives $...
0
votes
1answer
137 views

What is the value of following limit?

Let $P$ be a polynomial in complex variable $z$ of degree $d$ i.e. $P(z)= a_d z^d+.....+a_1 z+a_0$ Now I want to calculate following limit $f(z) = \limsup_{n \to \infty} \frac{1}{d^n} (Log|P(z)^{*...
18
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1answer
712 views

Poincaré metric on the Riemann sphere minus more than two points

If we omit more than two points from the Riemann sphere, we will obtain a hyperbolic Riemann surface endowed with a canonical metric descending from its universal cover which is the Poincaré disk. Let ...
4
votes
1answer
181 views

Julia set containing smooth curve

I have two realted questions. Let $R$ be a rational function on $\mathbb{C}$ with degree at least 2. We denote by $\mu$ the measure of maximal entropy for $R$ and recall that the Julia set coincides ...
2
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0answers
91 views

Is the Mandelbrot set weakly self-similar?

A subset $F$ of an Euclidean space $E$ will be called weakly self-similar if for all $x \in F$ there is $\epsilon_x>0$ such that for all positive $\epsilon \le \epsilon_x$ there are $y \in F$, $\...
4
votes
2answers
249 views

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 $P$?
14
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3answers
976 views

Convergence of Newton's method

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 $z_0$ ...
3
votes
0answers
59 views

Is a domain of a holomorphic flow pseudoconvex?

Let $Z$ be a holomorphic vector field on $\mathbb{C}^n$. I would like to know whether (it seems that it is) the domain $D_\phi \subset \mathbb{C} \times \mathbb{C}^n$ of a maximal flow $\phi: D_\phi \...
1
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0answers
91 views

Is $\partial M_d$ continuously determined by $d$?

This question is inspired by a question on math.stackexchange: https://math.stackexchange.com/questions/1707291/is-the-generalized-mandelbrot-set-a-fractal-in-the-d-dimension/2575089 The animation ...
2
votes
1answer
57 views

Are the immediate basin of these exponential maps simply connected?

This is a simple question that I direfully need an answer for. If the response is in the negative, I can work with it. If the response is in the positive, I can also work with it. I just can't seem to ...
3
votes
1answer
152 views

Is it known that MLC is sufficient to prove the density of hyperbolic conjecture of rational maps (or not)

Is it known that local connectivity of the Mandelbrot set (MLC) is sufficient prove the density of hyperbolic conjecture of qudratic family. I wondered is it known that the MLC is not enough (or ...
1
vote
1answer
84 views

Does the boundary of immediate basin contain a fixed point?

Let $f$ be a rational map of degree $d\geq 2$, and $B$ is a simply connected immediate basin of an supper-attracting fixed point of $f$. I want to know whether there exists a fixed point of $f$ ...
1
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1answer
83 views

Reference Request: Siegel Center Problem

Does anyone have a reference to where I may find a statement of the problem and perhaps (but not required) some elementary dicussion of Siegel's Center Problem?
1
vote
1answer
142 views

Hölder continuity of holomorphic motions

Let $D$ denote the complex unit disk and $X \subset \mathbb{C}$ some subset. Let us consider a holomorphic motion $i \colon D \times X \rightarrow \mathbb{C}$ (denoted $i_\lambda(z)$) meaning for each ...
4
votes
1answer
103 views

Can iterates of a non-polynomial function be bounded by an exponential indefinitely?

Assume $f$ is an entire non-polynomial function of arbitrarily small exponential order ('zero'th order' if you're into calling it that). Is it possible that for all $n$ we have $$|f^{\circ n}(z)| <...
1
vote
1answer
71 views

Finding the “orthogonal” map of a given 1d map

Let $f:\mathbb{C}\to\mathbb{C}$ be meromorphic or even entire. Let $z:\mathbb{C}\to\mathbb{C}$ be such that it holds $$z(t+1) = f(z(t)).$$ If $f$ is entire and we choose carefully $z$ can be ...
1
vote
1answer
138 views

Is there an example with Area $0<F(f)<\infty$ for some transcendental entire function

It seems that there may be example of a transcendental entire function with finite (but positive) planar area of the Fatou set in Eremenko-Lyubich class. However, I can't not find it in the ...
0
votes
1answer
177 views

Inverse image of a Jordan curve

If there exists a rational map $R$ from the extended plane $\hat{\mathbb{C}}$ to itself, and a Jordan curve $J$ on the plane, such that $R$ has no critical value on the curve, can we say that the ...