All Questions
Tagged with complex-dynamics ds.dynamical-systems
88 questions
1
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38
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Is there an equivalent to the logistic map for a nonlinear path through some of the other nodules of the Mandelbrot set?
The logistic map can be related to the real axis of the Mandelbrot set, looking at the different cycle lengths as you pass through all the various nodules along the real axis. But there are other ...
5
votes
1
answer
272
views
Why "no wandering domain" fails in parabolic basin?
Theorem (Sullivan). Every Fatou component $U$ of $f$ rational map is eventually periodic, that is, there exist $n > m > 0$ such that $f^n(U) = f^m(U)$
I am familiar with the proof: spread around ...
0
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0
answers
79
views
Alternative proof of parabolic implosion
I am working on an alternative proof of parabolic implosion from complex dynamics, but only allowing hyperbolic perturbation.
Theorem (Parabolic Implosion) Let $f(z)=z^2+z$ and $U_f$ be parabolic ...
1
vote
1
answer
133
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Does this sequence of Blaschke Product have rescaling limit $z-1$?
Background: The conformal conjugacy class of parabolic isometry of upper half plane $\mathbb{H}$ consists of $f(z) = z+1$ and $g(z)=z-1$.
Consider surjective proper holomorphic $F_n: \mathbb{H} \...
3
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1
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127
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Can doubly parabolic Blaschke product (BP) contained in another doubly parabolic BP?
Let $f:\mathbb{D}\rightarrow\mathbb{D}$ be a degree $d$ doubly parabolic Blaschke product with Denjoy-Wolff point at $z=1$. That is, $f(1) = 1$, $f'(1)=1$ and $f''(1)=0$.
Let $U \subset \mathbb{D}$ be ...
5
votes
1
answer
548
views
Connected set in a filled Julia set
Let $K$ be the filled Julia set of a complex polynomial of degree at least 2. Suppose that $K$ is connected. Let $p_1, \dots, p_N \in K$ be some points. Does there exist a connected set $K_N$ ...
- 303
1
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1
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153
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Accessible points of a simply connected domain
We know that if $U$ is an open subset of $\mathbb{\widehat C}$ (extended complex plane), a point $v\in\partial U$ is called accessible from $U$ if there exists a curve $\gamma:[0,1)\to U$ such that $\...
2
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1
answer
148
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Entire function of finite order with deficient value
There are some sufficient conditions for an entire function to have a finite deficient value e.g., if the order $\rho$ of an entire function $f$ is such that $2<\rho<+\infty$ with all but ...
1
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0
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71
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Multiply connected Fatou component of an entire function
This question may be trivial but still I want to know the answer.
Question: Is there any necessary condition (except boundedness of the Fatou component) for the existence of a multiply connected Fatou ...
2
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1
answer
119
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Any theorem shows that flowmap $\phi_{\sum_{i=1}^n a_i f_i(x)}^\tau$ can be approximated by $\phi_{f_{\theta(t)}(x)}^{\tau'}$?
Given a control family $F:=\{f_1,\dotsc,f_n\}$, and $\phi_f^\tau(x)$ is the flowmap of the dynamical system
$$
\begin{cases}
z'(t)=f(z),\\
z(0)=x,
\end{cases}
$$ at end time point $\tau$.
Suppose $a_i&...
- 109
2
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1
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187
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How to find the hyperbolic dimension of map $f(z) = z^2$ of $\overline{\mathbb{C}}$ onto itself?
I am reading the research article "The Hausdorff dimension of the boundary of Mandelbrot set and Julia sets" by Shishikura. In his article, he defined hyperbolic sets and hyperbolic ...
- 45
5
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1
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321
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What is the logical complexity of the Mandelbrot Local Connectivity conjecture? (Is it equivalent to a statement of arithmetic?)
Denote by MLC the statement “the Mandelbrot set is locally connected” and MHC the statement “hyperbolic components are dense in the Mandelbrot set” (it is known that MLC implies MHC, and whether ...
- 32.4k
2
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1
answer
120
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Examples of hyperbolic set and J-stable sets
I am reading the research article "The Hausdorff dimension of the boundary of Mandelbrot set and Julia sets" by Shishikura. The following two definitions are given without any examples in ...
- 45
4
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2
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419
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Borel summation and the Abel function of $e^z-1$
This is a question that has bothered myself and Gottfried Helms a fair amount of late. He has made his case for the following result, but a proof escapes both of us. The question is deceptively simple,...
- 388
2
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1
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174
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Finding the repelling fixed point of an exponential, knowing only its attracting one
This question has been bugging me for a while, I have an answer that is working sufficiently for the program I'm using, but it is a tad slow, and let's say imprecise. It is not an overtly difficult ...
- 388
1
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0
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61
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Holomorphic dynamical systems defined on a contractible bounded open subset of $\Bbb{C}^n$
Let $U$ be a contractible bounded open subset of $\Bbb{C}$. There is a standard classification of possible dynamical behaviors of holomorphic maps $f:U\rightarrow U$:
Attracting Case: There is an ...
- 3,599
18
votes
2
answers
2k
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Renormalization in physics vs. dynamical systems
I am studying complex dynamics, so to me renormalization of a dynamical system means something like a rescaled first-return map on (a subset of) the underlying space. I understand that in quantum ...
- 181
1
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0
answers
61
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Lower semicontinuity of the number of attracting periodic points of a holomorphic family of rational maps?
Recently I have been reading the book Mathematical Tools for One-Dimensional Dynamics.
In the proof of the theorem 5.4.2, authors use the following fact that the number of attracting periodic points ...
- 41
1
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1
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486
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Mandelbrot set and logistic map connection
I'm currently writing an undergraduate thesis on chaos theory with a particular focus on the connection between the Mandelbrot set and the logistic map. I have found scattered posts on this site, ...
3
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0
answers
80
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Confusion on the assumption when discussing the kneading invariants for unimodal maps
A unimodal map is a continuous map $f:[0,1]\longrightarrow [0,1]$ such that there is only one turning point (critical point), denoted by $c$, and $f(0)=f(1)=0$.
Unimodal map is related to kneading ...
- 131
4
votes
2
answers
332
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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\...
- 3,317
3
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3
answers
257
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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(...
- 3,317
5
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3
answers
363
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$ ...
- 26k
7
votes
1
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631
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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 ...
user153571
0
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1
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206
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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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1
answer
164
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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 $...
- 3,317
12
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2
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750
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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
$$
which defines a ...
- 32.4k
5
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0
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230
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Showing that a certain level set of a continuous family of holomorphic maps is locally path connected
I'm working with a continuous function $P: [0,1] \times W \to \mathbb{C}^n$, where $W \subset \mathbb{C}^n$ is an open, relatively compact ball centred at the origin. The map $P$ satisfies the ...
- 51
1
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0
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472
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The mysterious numbers $ \frac{13}{20} $ and $20$?
Let $g(x) = x^6 - 30 x $
Let $h(x) = x^6 $
Let $f(x) = x^2 - 2 $
Let $r$ be a reduced fraction $0 < \frac{p}{q} < 2 $ with integers $p,q > 1$
Let $f_{n+1}(x) = f(f_n(x)) = f_n(f(x)) , ...
- 769
13
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1
answer
452
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Is the set of escaping endpoints for $e^z-2$ completely metrizable?
Let $f:\mathbb C \to \mathbb C$ be the complex exponential $$f(z)=e^z-2.$$ It is known that $J(f)$, the Julia set of $f$, is a uncountable collection of disjoint rays (one-to-one continuous images ...
- 3,317
5
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1
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302
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An entire function all whose forward orbits are bounded
Edit: I revise the question according to the comment of Gabe Conant.
What is an example of a non constant entire function $f:\mathbb{C}\to \mathbb{C}$ which satisfy the following?:
For every $...
- 356
7
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2
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572
views
Smooth Julia set for quadratic polynomials
This question is related to a classification of rational maps in terms of properties of their Julia set.
Let $f= z^2 + c$, with $c\in \mathbb{C}$ be a quadratic polynomial such that its Julia set $J(...
- 303
1
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1
answer
160
views
A singular holomorphic foliation of $\mathbb{C}^2$ with a bounded leaf
Is there a polynomial vector field on $\mathbb{C}^2$ which possesses a bounded regular leaf? By bounded, I mean a bounded subset of $\mathbb{C}^2$.
- 356
3
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1
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162
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Symmetries for Julia sets of perturbations of polynomial maps
This is a naive question. Consider the
Julia sets
of the map
$$ z \mapsto z^n + \lambda / z^k $$
with $z,\lambda \in \mathbb{C}$, and the exponents $n,k \in \mathbb{N}$.
For example, for $n=k=3$, ...
- 151k
1
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1
answer
445
views
Formal group law and Koenigs function conjecture?
Let $f(x,y)$ be a symmetric real function and a formal group law
$$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 ...
- 769
0
votes
1
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275
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 ...
user135767
2
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0
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108
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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 ...
- 533
4
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0
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93
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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 $...
- 384
3
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0
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149
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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
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2
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336
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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$?
- 18.7k
1
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0
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100
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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 ...
- 709
4
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1
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116
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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)| <...
user78249
1
vote
1
answer
73
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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 ...
- 503
6
votes
0
answers
223
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Reference request: Complex geodesic flow
Can someone suggest a book on complex geodesic flow? I am interested in it mainly because I was told these form a very useful class of Riemann surface laminations. Of special interest to me is the ...
2
votes
2
answers
819
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Definition of Post-critically finite map
I'm studying the dynamic of the post-critically finite for my master thesis and my professor gave me the problem concerning generalization of post-critically finite ration map. Concretely, let $f : \...
- 293
5
votes
0
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309
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Is the closed orbit of the Van der Pol equation a stable periodic orbit?
We consider the Van der Pol vector field $$(1) \;\;\;\;\;\; \begin{cases} x'=y-(x^3-x)\\ y'=-x\end{cases}$$ on $\mathbb{R}^2.$
It is well known that this equation has a unique limit ...
- 356
38
votes
1
answer
3k
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Is the area of the Mandelbrot provably computable?
Recall the Mandelbrot set $M$ is the set of points $c$ in the complex plane such that the sequence $z_0 = 0, z_{n+1} = z_n^2 + c$ is bounded. It is well-known that $M$ is a compact set of positive ...
- 6,287
7
votes
3
answers
458
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A question about Julia set for quadratic family
Let $P_{c}(z)=z^2+c$. It seems from the software that the map between the parameter $c$ and the Julia set $J(P_c)$ is an injective map. Is there some reference about it? Any comments and reference ...
- 1,706
44
votes
4
answers
8k
views
Did Gaston Julia ever get to see a computer-generated image of his eponymous set?
I learned from Wikipedia that Gaston Julia died in 1978. Is it known if he ever got to see a computer-generated image of the set named after him?
- 457
1
vote
0
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134
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Inverse limits of the interval with a single bonding map below the identity
My question is as follows.
QUESTION. Is there a topological description of the class of arc-like continua that arise as inverse limits of $[0,1]$ with a single continuous surjective bonding map $f\...
- 6,548