All Questions
Tagged with complex-dynamics ds.dynamical-systems
88 questions
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 ...
- 91.8k
1
vote
0
answers
38
views
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 ...
- 63.9k
0
votes
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 ...
- 63.9k
6
votes
5
answers
2k
views
Precise location of the Mandelbrot Bulb Attachment to the main Cardioid
Is there an analytical formula for determining the location of the attachment points of the bulbs on the main cardioid? I was told there is an exact parametrization of the boundary of the main ...
1
vote
1
answer
133
views
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
votes
1
answer
127
views
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 $K_N$ ...
- 63.9k
1
vote
1
answer
153
views
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 $\...
CommunityBot
- 1
2
votes
1
answer
148
views
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
vote
0
answers
71
views
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
votes
1
answer
119
views
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&...
- 128k
5
votes
0
answers
309
views
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 ...
- 3,065
6
votes
4
answers
763
views
A follow up question related to entropy
For a self-map $\varphi:X\longrightarrow X$ of a space $X$, many important notions of entropy are defined through a limit of the form $$\lim_{n\rightarrow\infty}\frac{1}{n}\log a_n,$$ where in each ...
- 1,446
2
votes
1
answer
187
views
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
votes
1
answer
321
views
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 ...
- 12.4k
2
votes
1
answer
120
views
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 ...
- 6,548
4
votes
1
answer
286
views
Power series expansion of the Koenigs function
Given a non-zero holomorphic function $f$ fixing $0$ which isn't a Mobius transform, the Koenigs function of $f$, which we'll call $h$, is the function which linearizes $f$ in the sense that
$$
h(f(z))...
- 4,713
4
votes
2
answers
419
views
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
17
votes
5
answers
2k
views
Arithmetic dynamics and dynamics on moduli spaces
The following question is more of a request for pointers to suitable literature on introductory material for arithmetic dynamics and dynamics on moduli spaces.
In my dissertation, I have been ...
- 4,713
2
votes
1
answer
174
views
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
9
votes
2
answers
1k
views
Is this a Julia set (and if so, for which function family is it the Julia set)?
Consider the function family given by $f_\lambda(z) = z - p_\lambda(z)/p_\lambda'(z)$ where $p_\lambda(z) = (z^2 - 1)(z - \lambda)$. Every attracting cycle and every rational neutral cycle of $f_\...
- 4,713
26
votes
7
answers
2k
views
If you were to axiomatize the notion of entropy
What are the axioms that a good notion of entropy must satisfy? Please note that I am not asking for the definitions of various types of entropy such as topological entropy or measure-theoretic ...
CommunityBot
- 1
1
vote
0
answers
61
views
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
views
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 ...
- 10.5k
1
vote
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}$$
This equation has many ...
- 2,784
1
vote
0
answers
61
views
Lower semicontinuity of the number of attracting periodic points of a holomorphic family of rational maps?
Recently I have been reading the book Mathematical Tools for One-Dimensional Dynamics.
In the proof of the theorem 5.4.2, authors use the following fact that the number of attracting periodic points ...
- 4,713
4
votes
2
answers
332
views
Exponential iterates of a complex number
Let $f:\mathbb C\to \mathbb C$ be defined by $f(z)=e^z-1$. Let $f^n$ denote the $n$-fold composition of $f$.
In my new paper Erdős space in Julia sets I show that $$Z:=\{z\in \mathbb C:\lvert\...
- 6,548
15
votes
8
answers
4k
views
Are there some original papers or books related to applications of algebraic topology and algebraic geometry in complex dynamic systems
Recently I have much interest in algebraic topology and algebraic geometry. I am a student of the field of complex dynamical systems. According to my knowledge, my friends told me that there are many ...
- 2,821
1
vote
1
answer
486
views
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, ...
- 91.8k
3
votes
0
answers
80
views
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
3
votes
3
answers
257
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(...
- 3,599
5
votes
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 $\mathbb C_\infty$ ...
CommunityBot
- 1
7
votes
1
answer
631
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 ...
- 91.8k
0
votes
1
answer
206
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 ...
- 91.8k
1
vote
1
answer
164
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 $...
CommunityBot
- 1
12
votes
2
answers
750
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 ...
- 111
13
votes
1
answer
452
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 ...
- 3,317
5
votes
0
answers
230
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 ...
- 63.9k
1
vote
0
answers
472
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)) , ...
- 769
5
votes
1
answer
302
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 $...
- 3,274
7
votes
2
answers
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(...
- 91.8k
1
vote
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 $\mathbb{C}^2$.
- 28.9k
3
votes
1
answer
162
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 $n=k=3$, ...
- 23.2k
0
votes
1
answer
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 ...
- 47.4k
2
votes
0
answers
108
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 ...
- 63.9k
4
votes
0
answers
93
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 $...
- 384
3
votes
0
answers
149
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
2
answers
336
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 $P$?
- 2,083
1
vote
0
answers
100
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 ...
- 63.9k
44
votes
3
answers
4k
views
When does iterating $z \mapsto z^2 + c$ have an exact solution?
If one iterates the map $z \mapsto z^2 + c$ there is obviously a simple formula for the sequence one gets if $c=0$. Less obviously, there is also a simple formula when $c = -2$ (use the identity $2 \...
- 63.9k