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Questions tagged [complex-dynamics]

Dynamics of holomorphic transformations; Mandelbrot and Julia sets.

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2 votes
0 answers
89 views

Conjugacy between piecewise linear circle maps

Let $\mathcal{M}$ the Mandelbrot set, $\mathcal{M}=\{c \in \mathbb{C}: \{Q_c^n(0) \}_{n \in \mathbb{N}} \text{ is bounded, where } Q_c(z)=z^2+c \}$ And let the hyperbolic or stable component, $H_n=\{ ...
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 ...
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 ...
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 ...
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 ...
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 ...
0 votes
1 answer
68 views

Looking for a precise statement about hyperbolic points in the interior of the Mandelbrot set

A Numberphile video piqued my interest regarding the hyperbolicity property of points in the Mandelbrot set. But I can't seem to find a concise statement about the conjecture about hyperbolic points ...
2 votes
1 answer
195 views

Dynamics of $e^z+z^2+z+1$

Question: How to find the dynamics of $f(z)=e^z+z^2+z+1$ also how to find the escaping set ? Since critical points control the dynamics, I want to find the critical points first, but I am not getting ...
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 $K_N$ ...
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
2 answers
2k views

Composition of circle inversions

I would like to understand the map of $\mathbb{C}$ to $\mathbb{C}$ that results by iterating inversion in a unit circle. Let $f(z)$ for $z \in \mathbb{C}$ invert $z$ in a unit circle centered on $q_1$,...
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(\{...
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 ...
11 votes
1 answer
427 views

Is the Mandelbrot set Suslinian?

The Mandelbrot set is known to be (path-)connected and compact. A non-degenerate space with these properties is called a continuum. A continuum $X$ is Suslinian if every collection of non-degenerate ...
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 $d$ ...
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&...
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 $...
19 votes
1 answer
4k views

Is the area of the Mandelbrot set known? [duplicate]

The Mandelbrot set has an area; is it known exactly? If so, how, and what is the value? If not, why is this a hard question to answer?
8 votes
1 answer
495 views

$f(f(z)) = z , f(\exp(z)) = \exp(f(z)) $?

While talking about tetration with my friend the following idea (re)occured. $$f(f(z)) = z ,\quad f(\exp(z)) = \exp(f(z)) \tag{A}\label{A}$$ or variations of it like the weaker $$f(f(f(f(z)))) = z ,\...
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 ...
2 votes
1 answer
118 views

Uniformization of Julia sets and lacunary series

If the complement of a Julia set of quadratic polynomial z^2+c is locally connected and simply connected, it is uniformized by the complement of the unit disk. Consider the uniformization map and its ...
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 (...
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. ...
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 ...
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. ...
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 ...
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 ...
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 ...
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 ...
8 votes
1 answer
766 views

Does the Mandelbrot set have dense interior?

Let $M$ be the Mandelbrot set. Question: Is the interior of $M$ dense in $M$? My intuition is that this is true, and moreover that hyperbolic components of the interior are dense in $M$ as well, and ...
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))...
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 ...
19 votes
2 answers
11k views

Meaning of $\Subset$ notation

The symbol $\Subset$ (occurring in places where $\subseteq$ could occur syntactically) comes up frequently in a paper I'm reading. The paper lives at the intersection of a few areas of math, and I ...
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,...
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 ...
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 ...
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 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{...
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_\...
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 ...
11 votes
0 answers
570 views

A curious observation on the elliptic curve $y^2=x^3+1$

Here is a calculation regarding the $2$-torsion points of the elliptic curve $y^2=x^3+1$ which looks really miraculous to me (the motivation comes at the end). Take a point of $y^2=x^3+1$ and ...
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 ...
9 votes
1 answer
349 views

Tiling the plane with finitely many congruent pieces

Suppose $A_1,\dots,A_n$ are measurable subsets of the plane that are all related by rigid motions such that $|(A_1 \cup \dots \cup A_n)^c| = 0$ and $|A_i \cap A_j| = 0$ for all $1 \leq i < j \leq n$...
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 ...
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 ...
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 ...
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 ...

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