All Questions
Tagged with cv.complex-variables complex-dynamics
82 questions
5
votes
1
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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
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 ...
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} \...
7
votes
2
answers
186
views
Non-locally connected polynomial Julia sets
What are some examples of complex polynomials whose Julia sets are connected, but not locally?
In the book Complex Dynamics by Carleson and Gamelin, I found:
They seem to reference:
But what is a ...
- 3,317
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 ...
11
votes
1
answer
959
views
Can the topologist's sine curve be realized as a Julia set?
Does there exist a rational function $f\in\Bbb{C}(z)$ whose Julia set coincides with
$$
T:=\left\{\left(x,\sin\left(\frac{1}{x}\right)\right)\,\Big|\,x\in\left(0,\frac{1}{\pi}\right]\right\}\cup\big(\{...
- 3,599
1
vote
1
answer
153
views
Accessible points of a simply connected domain
We know that if $U$ is an open subset of $\mathbb{\widehat C}$ (extended complex plane), a point $v\in\partial U$ is called accessible from $U$ if there exists a curve $\gamma:[0,1)\to U$ such that $\...
2
votes
1
answer
148
views
Entire function of finite order with deficient value
There are some sufficient conditions for an entire function to have a finite deficient value e.g., if the order $\rho$ of an entire function $f$ is such that $2<\rho<+\infty$ with all but ...
3
votes
1
answer
119
views
The number of components of the preimage of a continuum for a polynomial
Given a polynomial f with degree d, we can define a dynamical system $(\mathbb{C}, f)$. If we have a proper continuum $N \subset \mathbb{C}$, it is known that the set $f^{-1}(N)$ has at most $d$ ...
- 97
1
vote
1
answer
100
views
Orbit closure of two elliptic Möbius transformations
Let $g_1$ and $g_2$ be two elliptic Möbius transformations of infinite order in $\mathrm{Aut}(\bar{\mathbb{C}})$. If $\mathrm{Fix} (g_1) \cap \mathrm{Fix} (g_2) = \emptyset$, then can we deduce that $...
- 661
2
votes
1
answer
115
views
Dense orbits for a rational map
Given a complex rational function $f$ and $z\in \mathbb C$, let $O^+(z)=\{f^n(z):n\geq 1\}$. Let $$D(f)=\big\{z\in \mathbb C:\overline{O^+(z)}=J(f)\big\}.$$
So $D(f)$ is the set of points whose (...
- 3,317
5
votes
1
answer
120
views
Jordan curve boundaries of Fatou components
Let $f:\mathbb C\to \mathbb C$ be a rational map and let $J(f)$ and $F(f)$ denote the Julia and Fatou sets of $f$, respectively.
Let $\mathcal S$ be the set of all boundaries of Fatou components. ...
- 3,317
2
votes
0
answers
89
views
Finding a branch cut or a branch point [closed]
Is there a way to find a branch cut or a branch point, through which a curve over a complex function goes, or in general in some region of complex function, say $\ln(f(z))$, using numerical methods or ...
- 67
4
votes
2
answers
419
views
Borel summation and the Abel function of $e^z-1$
This is a question that has bothered myself and Gottfried Helms a fair amount of late. He has made his case for the following result, but a proof escapes both of us. The question is deceptively simple,...
- 388
2
votes
1
answer
174
views
Finding the repelling fixed point of an exponential, knowing only its attracting one
This question has been bugging me for a while, I have an answer that is working sufficiently for the program I'm using, but it is a tad slow, and let's say imprecise. It is not an overtly difficult ...
- 388
10
votes
1
answer
705
views
On entire functions with polynomial Schwarzian derivative
The Schwarzian derivative of an entire holomorphic function $f$ is defined as
$$Sf:=\left(\frac{f^{''}}{f'}\right)'-\frac{1}{2}\left(\frac{f^{''}}{f'}\right)^2.$$
In the following, we only consider ...
- 1,350
1
vote
0
answers
80
views
Positive integration on P^1
Let $u: \mathbb{P}^1(\mathbb{C}) \longrightarrow \mathbb{R}$ be a smooth function s.t. $u$ is invariant under complex conjugation and $\displaystyle \int_{\mathbb{P}^1(\mathbb{C})}u \; \omega_{\mathrm{...
- 171
3
votes
1
answer
166
views
A question about decompositions of rational functions
Let $f_1,g_1,f_2,g_2$ be non-constant rational functions on the Riemann sphere (i.e. elements of $\Bbb{C}(z)-\Bbb{C}$) satisfying $f_1\circ g_1=f_2\circ g_2$. Suppose there is a prime number $p$ such ...
- 3,599
4
votes
1
answer
428
views
Ahlfors' proof of Bloch's theorem
In his pioneering paper An extension of Schwarz's lemma, Ahlfors proves the lower bound on the Bloch constant $B \geq \frac{\sqrt{3}}{4}$. The proof of this lower bound proceeds as follows:
Let $W$ be ...
- 137
2
votes
0
answers
305
views
Understanding a more intricate Schwarz reflection principle--A question about Tetration
everyone. This is going to be a long question as it requires a good amount of back story in theory. This question is mostly along the lines: "I think this should happen, and I think my proof is ...
- 388
10
votes
0
answers
303
views
the (non-existent) group of conformal transformations
In physics intros to 2d conformal field theory, people often talk about the "group of conformal transformations". Of course, that's not a group but rather a pseudo-group... that's not what ...
- 43.2k
16
votes
3
answers
1k
views
If I have zeros at the vertices of an icosahedron, where should the poles go?
I've been tinkering with Newton's method applied to polynomials. E.g., Newton's method for $z^5 - 1 = 0$ gives:
There aren't a lot of symmetric patterns of finite sets of points in the plane, so I ...
- 1,788
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 ...
- 41
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, ...
2
votes
0
answers
55
views
Integral curves of rational vector fields and approximations
The following is the formal statement of a conjecture that feels almost obvious, but I cannot find a reference for it. The idea is that one can obtain the integral curves of a vector field $V(z)$ by ...
- 15.8k
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\...
- 3,317
0
votes
0
answers
104
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 ...
- 101
10
votes
1
answer
419
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 ...
- 15.8k
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(...
- 3,317
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 $\mathbb C_\infty$ ...
- 26k
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 ...
user153571
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 ...
- 32.4k
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 ...
- 51
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 $...
- 356
3
votes
0
answers
84
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 ...
- 131
4
votes
1
answer
127
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$ ...
- 388
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 ...
- 533
7
votes
2
answers
358
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 ...
- 589
4
votes
0
answers
73
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 \...
- 437
1
vote
1
answer
110
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 $f$ ...
- 143
1
vote
1
answer
196
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 ...
- 257
4
votes
1
answer
116
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
$$|f^{\circ n}(z)| <...
user78249
1
vote
1
answer
197
views
Inverse image of a Jordan curve
If there exists a rational map $R$ from the extended plane $\hat{\mathbb{C}}$ to itself, and a Jordan curve $J$ on the plane, such that $R$ has no critical value on the curve, can we say that the ...
- 11
1
vote
0
answers
294
views
Can an entire function have every root function?
My question is an amalgamation of two previous questions. The first question I'd like to draw attention to is here. It asks whether there can exist a non trivial semigroup defined on $\mathbb{C}$
$$\...
user78249
7
votes
1
answer
248
views
Are there such things as non-trivial entire semigroups?
I'll state the theorem I am posing up front, and then explain why I think this theorem appears to be true. I am asking if anyone can prove it, or knows references to where it is proved. Please, ...
user78249
1
vote
0
answers
52
views
Composing between Schröder functions in complex dynamics
Assume that $f(z)$ is a holomorphic function that sends some open and connected set $G$ to itself. Assume $f$ has a single fixed point $z_0$. Assume $f(f(...(n\,times)...f(z))) = f^{\circ n}(z) \to ...
user78249
2
votes
1
answer
763
views
What about the other $f$ such that $f(f(x)) = \sin(x)$?
This question is inspired by the big MO question here; and also inspired by the big MO question here. The premise of this question requires a backdrop on fractional iteration; so I'll start slow.
...
user78249
1
vote
1
answer
379
views
Infinite compositions of holomorphic functions, is there literature on the subject?
I've recently become very intrigued by infinite compositions. To get at what I mean by the term, I'll be as explanatory as possible.
Consider a sequence of holomorphic functions $\{\phi_j\}_{j=0}^\...
user78249
4
votes
1
answer
206
views
Super attracting fixed points have no fractional iteration
My question is really easy to state, but I'm having trouble hitting the final nail in the coffin in a proof of the result. The question concerns fractional iterations of holomorphic functions, for ...
user78249
1
vote
0
answers
107
views
Effective estimates for circle packing
The Riemann map from a simply connected domain to the unit disc can be approximated by circle packings thanks to a theorem of Rodin and Sullivan. (That is, take smaller and smaller triangulations and ...
- 3,373