Questions tagged [complex-dynamics]

Dynamics of holomorphic transformations; Mandelbrot and Julia sets.

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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?
T. Donaldson's user avatar
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 \...
Richard Borcherds's user avatar
37 votes
1 answer
3k views

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 ...
Jason Rute's user avatar
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35 votes
3 answers
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The deep significance of the question of the Mandelbrot set's local connectedness?

I am given to understand that the celebrated open problem (MLC) of the Mandelbrot set's local connectness has broader and deeper significance deeper than some mere curiosity of point-set topology. ...
David Feldman's user avatar
32 votes
3 answers
2k views

How is the Julia set of $fg$ related to the Julia set of $gf$?

Let $f$ and $g$ be complex rational functions (of degree $\geq 2$ if that helps). What can be said about the relationship between $J(fg)$ and $J(gf)$, the Julia sets of the composite functions $f \...
Tom Leinster's user avatar
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27 votes
5 answers
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Why are the Julia sets so simple? (quadratic family)

I want to know why, when I look at the Julia sets of the quadratic family, I see only a finite number of repeating patterns, rather than a countable infinity of them. My question is specifically ...
Andrea's user avatar
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27 votes
3 answers
936 views

A point set of power series with coefficients in {-1, 1}. Connected or not?

Let $z$ be a fixed complex number with $|z|<1$ and consider the set $$X_z := \Big\{\sum\limits_{i=1}^{\infty} a_i z^i \ \Big|\ a_i\in \{-1,1\} \forall i\Big\}.$$ What can be said about the set $M$ ...
Kirby Lee's user avatar
  • 373
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 ...
Mahdi Majidi-Zolbanin's user avatar
25 votes
6 answers
5k views

Parametrization of the boundary of the Mandelbrot set

Does anyone know how to parametrize the boundary of the Mandelbrot set? I am not a fractal-geometer or a dynamical systems person. I just have some idle curiosity about this question. The ...
David Richter's user avatar
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
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24 votes
1 answer
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Anti-Mandelbrot set

I clearly remember seeing a paper where the dynamic of the anti-conformal map $f(z)=\overline{z}^2+c$ was studied (the bar means complex conjugation). There was a picture of the analog of the ...
Alexandre Eremenko's user avatar
22 votes
4 answers
984 views

Representing a number close to 1 with a sum of reciprocals of natural numbers

For positive integers $n_1, \ldots, n_k$, let $H(n_1, \ldots, n_k)$ denote $1/n_1 + \ldots + 1/n_k$. Let $V(N)$ be the largest possible value of $H(n_1, \ldots, n_k)$ that is less than 1, subject to ...
Jeremy Kahn's user avatar
21 votes
1 answer
886 views

Is there a reference for "computing $\pi$" using external rays of the Mandelbrot set?

I was recently reminded of the following cute fact which I will state as a proposition to fix notation: Proposition Given $\epsilon > 0$, let $c = -3/4 + \epsilon i \in \mathbb{C}$ and $q_c(z) = z^...
Oliver Nash's user avatar
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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
18 votes
2 answers
10k 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 ...
Linda Brown Westrick's user avatar
18 votes
1 answer
894 views

Poincaré metric on the Riemann sphere minus more than two points

If we omit more than two points from the Riemann sphere, we will obtain a hyperbolic Riemann surface endowed with a canonical metric descending from its universal cover which is the Poincaré disk. Let ...
Amin Talebi's user avatar
18 votes
2 answers
3k views

Is there some known way to create the Mandelbrot set (the boundary), with an iterated function system?

Is there some known way to create the Mandelbrot set (the boundary), with an iterated function system (IFS)? Julia sets can be formed by iterating the two functions $z \mapsto \pm \sqrt{z-c},$ and ...
Per Alexandersson's user avatar
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
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 ...
user avatar
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
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 ...
14 votes
3 answers
1k views

Convergence of Newton's method

For a polynomial $P$ of degree $n$ with real coefficients and with $n$ distinct real roots, the Newton's method $z_{n+1} = z_n - {P(z_n) \over P'(z_n)}$ converges for almost all initial values $z_0$ ...
coudy's user avatar
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13 votes
1 answer
451 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 ...
D.S. Lipham's user avatar
  • 3,045
13 votes
0 answers
747 views

Algebraic proofs of algebraic theorems about algebraically closed fields

It is well-known that the first order theory of algebraically closed fields admits quantifier elimination, whence the theory $ACF_p$ of algebraically closed fields of given characteristic $p$ is ...
Adam Epstein's user avatar
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13 votes
0 answers
305 views

Diophantine approximation in the Julia set

Let $f : \mathbb{CP}^1 \to \mathbb{CP}^1$ be a rational map of degree $q > 1$; or just a quadratic binomial $z^2 + c$, if one prefers. The Julia set $J_f$ is the closure of the repelling periodic ...
Vesselin Dimitrov's user avatar
12 votes
4 answers
969 views

Rounding errors in images of Julia sets

One typically computes Julia sets by iterating a complex function, such as a polynomial or rational function. How do rounding errors affect the results? I'm looking for references on this issue, ...
lhf's user avatar
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12 votes
3 answers
384 views

Dynamics in one matrix variable

Are dynamical systems $$X \mapsto F(X)$$ studied where $X \in \mathrm{M}_n$, $\mathrm{M}_n:=\mathrm{Mat}(n,\mathbb{C})$ or $\mathrm{Mat}(n,\mathbb{R})$, and $F$ is a (properly defined noncommutative)...
Qfwfq's user avatar
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12 votes
2 answers
712 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 ...
Gro-Tsen's user avatar
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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
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
10 votes
4 answers
721 views

When are those subgroups of $\mathrm{SL}(2, \mathbb{C})$ discrete?

Let $A = \pmatrix{1 & 0 \\ \alpha & 1} $ and $ B = \pmatrix{1 & 1 \\ 0 & 1}$, where $\alpha \in \mathbb{C}$ is a complex parameter. Now consider the family of representations $r_{\...
Selim G's user avatar
  • 2,626
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
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
10 votes
1 answer
356 views

On the conformal removability of Jordan curves

We say that a compact subset $E$ of the Riemann sphere $\mathbb{C}_\infty$ is (conformally) removable if every homeomorphism of $\mathbb{C}_\infty$ conformal outside $E$ is actually conformal ...
Malik Younsi's user avatar
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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
10 votes
0 answers
526 views

What is the "category of bifurcations"?

While reading the introduction to this paper by Curtis McMullen, I came to the following (bold added): In this paper we show that every bifurcation set contains a copy of the boundary of the ...
Darsh Ranjan's user avatar
  • 5,912
9 votes
6 answers
1k views

When does the sequence of iterates of a rational function converge?

Darsh asks at the 20-questions seminar: Let $f:P^1 \rightarrow P^1$ be rational function. Can you say when the sequence $\{ f^n(x)\}_n=\{ x,f(x),f(f(x)),\cdots\} $ converges? What about the sequence ...
20 questions's user avatar
  • 1,029
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_\...
Aaron Golden's user avatar
9 votes
2 answers
804 views

How many times line segments can intersect a Jordan curve?

I posted a question on math.stackexchange.com but it seems this question might be open https://math.stackexchange.com/questions/109752/line-segments-intersecting-jordan-curve Namely, is there a set ...
Jaakko Seppälä's user avatar
9 votes
2 answers
1k views

Dynamics of Riemann zeta function

Has the dynamics of the Riemann zeta function been studied? By dynamics I mean the limiting behavior of the sequence of iterates $s, \zeta (s), \zeta (\zeta (s)), \zeta (\zeta (\zeta (s)))\dots $ for ...
user137686's user avatar
9 votes
2 answers
624 views

Periodicity in iterated powers of sin, cos, exp

Given a complex number $z$, consider the sequence \begin{align*} a_0 & = 1\\ a_1 & = (cos(1))^z\\ a_n & = (cos(a_{n-1}))^z \end{align*} This question is about trying to understand ...
Niles's user avatar
  • 589
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
9 votes
1 answer
516 views

When is a Newton basin fractal continuously determined by the roots of its polynomial?

Newton basin fractals are visualizations of the Julia sets of functions of the form: $$f_p(z) = z - p(z)/p'(z)$$ where $p$ is a complex polynomial. My question is: When is the Julia set, $J(f_p)$...
Aaron Golden'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
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
8 votes
3 answers
2k views

Harmonic level sets and boundary data

This is probably a classic problem, so a good reference book or paper to get me started on this type of question would be great: Let $\mathbb{D} \subset \mathbb{C}$ be the unit disk with boundary $\...
partition_of_unity's user avatar
8 votes
2 answers
381 views

Linearizing a power series by conjugation

Let $\mathfrak{I}:=\big\{ \, f:=\sum_{k=0}^\infty f_k z^k \in\mathbb{C}[[z]]\; : \text{s.t. }\; f_0=0 \;\text{ and }\; f_1=1\big\}$. A most basic result about linearization states that, for any $f\...
Pietro Majer's user avatar
  • 56.3k
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
7 votes
3 answers
448 views

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 ...
yaoxiao's user avatar
  • 1,654
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 ...
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