Questions tagged [complex-dynamics]

Dynamics of holomorphic transformations; Mandelbrot and Julia sets.

Filter by
Sorted by
Tagged with
0 votes
1 answer
74 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 $\...
Factorial_zero's user avatar
2 votes
1 answer
133 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 ...
Factorial_zero's user avatar
1 vote
0 answers
56 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 ...
Factorial_zero's user avatar
3 votes
1 answer
111 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 $d$ ...
Yee Neil's user avatar
11 votes
1 answer
348 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 ...
D.S. Lipham's user avatar
  • 3,045
2 votes
1 answer
112 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&...
li ang Duan's user avatar
1 vote
1 answer
91 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 $...
Chicken feed's user avatar
19 votes
1 answer
3k 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?
user6873235's user avatar
1 vote
1 answer
101 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 (...
D.S. Lipham's user avatar
  • 3,045
2 votes
1 answer
107 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 ...
0x11111's user avatar
  • 473
5 votes
1 answer
106 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. ...
D.S. Lipham's user avatar
  • 3,045
2 votes
0 answers
82 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 ...
roignoirewg's user avatar
2 votes
1 answer
180 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 ...
N_Set's user avatar
  • 45
0 votes
0 answers
78 views

Persistence of irrationally indifferent periodic points

I am trying to see when an irrationally indifferent periodic point persists for a holomorphic family of rational maps on the Riemann sphere. This is a question I am currently pondering after reading ...
Maths Matador's user avatar
8 votes
1 answer
479 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 ,\...
mick's user avatar
  • 721
5 votes
1 answer
273 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 ...
Gro-Tsen's user avatar
  • 29.8k
2 votes
1 answer
87 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 ...
N_Set's user avatar
  • 45
8 votes
1 answer
712 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 ...
Geoffrey Irving's user avatar
4 votes
2 answers
387 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,...
Richard Diagram's user avatar
2 votes
1 answer
162 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 ...
Richard Diagram's user avatar
10 votes
1 answer
674 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 ...
student's user avatar
  • 1,320
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{...
QU Binggang's user avatar
11 votes
0 answers
497 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 ...
KhashF's user avatar
  • 2,588
3 votes
1 answer
160 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 ...
KhashF's user avatar
  • 2,588
9 votes
1 answer
347 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 $1 \leq i < j \leq n$...
James Propp's user avatar
  • 19.4k
4 votes
1 answer
409 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 ...
AshyK's user avatar
  • 137
2 votes
0 answers
256 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 ...
Richard Diagram's user avatar
10 votes
0 answers
300 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 ...
André Henriques's user avatar
1 vote
0 answers
60 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 ...
KhashF's user avatar
  • 2,588
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 ...
CAT in hat's user avatar
2 votes
0 answers
73 views

When is replacing the prefix of an angled internal address a valid operation?

While working on an artwork exploring patterns in the Mandelbrot set fractal, I constructed an angled internal address by: $$ 1 \overset{1/2}\longrightarrow 2 \overset{1/2}\longrightarrow 3 \overset{1/...
Claude's user avatar
  • 101
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 ...
Geoffrey Irving's user avatar
0 votes
1 answer
351 views

On the relevance of the property $\exp^{\circ a}(\exp^{\circ b}(z))=\exp^{\circ a+b}(z)$ for the *fractional* iteration ("tetration")

In the concept of fractional iteration of the exponential function ("tetration") the property of $$\exp^{\circ a}(\exp^{\circ b}(z))=\exp^{\circ b}(\exp^{\circ a}(z))=\exp^{\circ a+b}(z) \...
Gottfried Helms's user avatar
4 votes
2 answers
362 views

Hausdorff dimension of Julia set

Can anyone show me the proof "Hausdorff dimension of Julia set is strictly positive"? For purpose to prove this we might have to prove the green function of basin of attraction to infinity ...
matthew's user avatar
  • 51
1 vote
0 answers
60 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 ...
CuriousTiger's user avatar
6 votes
2 answers
237 views

Cutting a Julia set into infinitely many pieces at finitely many points

Let $f\colon \widehat{\mathbb{C}}\to \widehat{\mathbb{C}}$ be a rational function of degree two or greater whose Julia set $J_f$ is connected. If $S\subseteq J_f$ is a finite set of periodic points, ...
Jim Belk's user avatar
  • 8,433
1 vote
0 answers
73 views

Clarification about the process of naturally endowing a space with a Riemann orbifold structure supported on a sphere

I am having some difficulties understanding an argument in a proof. Here is an excerpt from Lyubich–Peters - Classification of invariant Fatou components for dissipative Henon maps, first geometric ...
nandi's user avatar
  • 53
1 vote
1 answer
455 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, ...
Person21312412's user avatar
2 votes
0 answers
49 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 ...
Per Alexandersson's user avatar
3 votes
0 answers
76 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 ...
JacobsonRadical's user avatar
9 votes
0 answers
310 views

Discriminants of Gleason's period-$n$ polynomials for the Mandelbrot set

Gleason's polynomials are the sequence of monic integer polynomials defined recursively by $$ \prod_{d \mid n} G_d(c) = (((c^2+c)^2+c)^2+\cdots+c)^2+c \quad \quad \quad [\textrm{$n$ iterates}], $$ for ...
Vesselin Dimitrov's user avatar
4 votes
1 answer
280 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 irreducible ...
Curiosity's user avatar
  • 293
4 votes
2 answers
321 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\...
D.S. Lipham's user avatar
  • 3,045
0 votes
0 answers
92 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 ...
Hilton's user avatar
  • 101
25 votes
2 answers
1k 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 ...
James Propp's user avatar
  • 19.4k
10 votes
1 answer
397 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 ...
Per Alexandersson's user avatar
3 votes
3 answers
253 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(...
D.S. Lipham's user avatar
  • 3,045
5 votes
3 answers
335 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 $\mathbb C_\infty$ ...
Joël's user avatar
  • 25.7k
7 votes
1 answer
611 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 ...
user avatar
0 votes
1 answer
198 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 ...
user152801's user avatar