Dynamics of holomorphic transformations; Mandelbrot and Julia sets.

**31**

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**3**answers

1k 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 \...

**27**

votes

**3**answers

816 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 $M$ ...

**23**

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 ...

**22**

votes

**3**answers

3k views

### 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.
...

**22**

votes

**1**answer

1k views

### 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 ...

**20**

votes

**4**answers

705 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 ...

**19**

votes

**6**answers

2k 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 ...

**17**

votes

**1**answer

392 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) = ...

**14**

votes

**5**answers

997 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 ...

**13**

votes

**2**answers

1k 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 ...

**12**

votes

**1**answer

5k 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 ...

**12**

votes

**0**answers

174 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 ...

**11**

votes

**3**answers

348 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)...

**10**

votes

**4**answers

490 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_{\...

**10**

votes

**4**answers

627 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, ...

**10**

votes

**1**answer

212 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 ...

**9**

votes

**8**answers

1k 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 field of complex dynamic systems. According to my knowledge, my friends told me that there are many ...

**9**

votes

**1**answer

372 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)...

**9**

votes

**0**answers

463 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 ...

**8**

votes

**2**answers

521 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
http://math.stackexchange.com/questions/109752/line-segments-intersecting-jordan-curve
Namely,
is there a set $...

**8**

votes

**2**answers

838 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_\...

**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 $\...

**8**

votes

**0**answers

396 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 ...

**7**

votes

**5**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 ...

**7**

votes

**2**answers

310 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 ...

**6**

votes

**3**answers

229 views

### Clustering of periodic points for a polynomial iteration of $\mathbb{C}$

Let $f : \mathbb{C} \to \mathbb{C}$ be a polynomial map of degree $q > 1$. Consider $E_n \subset \mathbb{C}$ the set of periodic points with period (dividing) $n$; generally, $|E_n| = q^n$. Since ...

**6**

votes

**3**answers

248 views

### Is there a effective computational criterion to all periodic points of a rational function are repelling.

I came up with a question to know the fatou component of of some types of rational function. In some sense, I may need to give a computational criterion to existence of attracting periodic basin for a ...

**6**

votes

**4**answers

632 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 ...

**6**

votes

**2**answers

670 views

### Symmetries of the Julia sets for $z^2+c$

The julia set seems to have symmetries roughly corresponding to translation, rotation and scaling.
In the following image
You can see the horizontal translation, which leaves the extremal left and ...

**6**

votes

**0**answers

227 views

### Is there an efficient way to visualize the bifurcation locus of this family of functions?

I have been trying to help out with this question from math.stackexchange. It concerns the family of functions:
$$f_\alpha(z,w) = \frac{\alpha + z}{1 + w}.$$
and an iteration scheme:
$$z_{n+1} = ...

**6**

votes

**0**answers

177 views

### Measure theoretic entropy

I don't know if this is an elementary question or not. In what follows all maps are continuous
Suppose that $P:\mathbb{C}\rightarrow\mathbb{C}$ is a complex polynomial of degree $d>1$ and let $\mu$...

**5**

votes

**2**answers

415 views

### Roadmap to Complex Dynamics (Particularly the works of Hubbard, Douady, and Yoccoz regarding the Mandelbrot set)

As others have had great success with their question, I hope to ask one in a similar vein. As a student who has some background in complex analysis and dynamical systems, I am hoping to explore ...

**5**

votes

**1**answer

491 views

### Fatou Coordinate for function with rationally indifferent fixed point, and repelling fixed point

Lets say I have $f(z)=z^2+c$, with $c=0.35676274578 + 0.32858194507i$. Then $f(z)$ has a fixed point $\kappa_0=0.15450849719 + 0.47552825815i$, which is rationally indifferent with a period $m=5$. ...

**4**

votes

**3**answers

504 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 ...

**4**

votes

**2**answers

705 views

### Is complex analytic extension of real-analytic diffeomorphism a diffeomorphism ?

Hi, my question is :
Let $D$ be the open unit disk in $\mathbb{C}=R^2$ and $f:D\to D$ be a real-analytic diffeomorphism. Let us think of the canonical embedding : $\mathbb{C}=R^2\subset \mathbb{C^2}. ...

**4**

votes

**2**answers

374 views

### Is there any elementary proof of No wandering domain for polynomials

It seems that it is almost impossible to give a elementary proof of Sullivan's no wandering domain for rational map or even more general class of maps.
I think it is interesting to ask whether we ...

**4**

votes

**2**answers

385 views

### Algebraicity of the “outer” boundary of the Mandelbrot set

Let $M$ be the Mandelbrot set and let $\lambda\in M, \mu\in \mathbb C$ be algebraic numbers. Let $t_{\lambda,\mu}$ be defined as
$$
t_{\lambda,\mu} = \sup \lbrace t\in \mathbb R\colon \lambda +t\mu \...

**4**

votes

**1**answer

65 views

### Integral Expression in Complex Dynamics

Let $\phi\in \mathbb{C}(z)$ be a degree $d\geq 2$ rational map, which we can write as $\phi = \frac{f}{g}$ for $f,g\in \mathbb{C}[z]$. Let $\omega_{FS}$ denote the Fubini-Study form on $\mathbb{P}^1(\...

**4**

votes

**0**answers

149 views

### Dynamical Mordell-Lang on Kahler manifolds?

Suppose that $X$ is a smooth projective variety over $\mathbb C$ and $\phi : X \to X$ is an endomorphism. Let $p \in V$ be a point and $V \subset X$ a subvariety. The dynamical Mordell-lang ...

**3**

votes

**3**answers

423 views

### Examples of cubic Julia sets

I'm looking for information on explicit cubic filled Julia sets with interesting properties such as:
Having two different finite attractors (such as $f(z)=z^3-1.5z$)
Being disconnected with non-...

**3**

votes

**1**answer

197 views

### Fixed points on Riemann surface

It is well known theorem that for a conformal mapping $\phi$ from a bounded and planar domain $\Omega$ to itself has three fixed points , then it must be identity mapping. However, I cannot find a ...

**3**

votes

**2**answers

390 views

### Hausdorff dimension of Julia sets of quadratics not in the Mandelbrot set.

What are the bounds on the possible values of the Hausdorff dimension of the Julia sets of quadratics not in the Mandelbrot set? In particular, assume we have a quadratic $q_c: z \mapsto z^2 + c$ on ...

**3**

votes

**3**answers

676 views

### Analytic ODE with complex time

Suppose we have a complex vector field on $\mathbb{C}^n$ which is analytic and has $|DV| < L$ on ball $B_r$ with radius r.
I would like to understand:
1) if there exists an analytic flow $\phi_t(x)...

**3**

votes

**1**answer

185 views

### Smoothness in Ecalle's method for fractional iterates

Some four years ago I answered my own question on fractional iteration, concluding that there is a half iterate of sine, that is $f(f(x)) = \sin x,$ which is real analytic for $0 < x < \pi$ but ...

**3**

votes

**1**answer

155 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))...

**3**

votes

**1**answer

125 views

### Reference for instability of Newton basins of polynomials at “separation” of a multiple root

In a previous question on MO I mentioned that I had convinced myself of the following:
When $f_p(z) = z - p(z)/p'(z)$ and $p$ is a complex polynomial, the Julia set, $J(f_p)$ is not continuously ...

**3**

votes

**1**answer

159 views

### Is there literature available on iterated function systems of the form $f^n = (g f^{n - 1}, g f^{n - 2}, \ldots)$?

This question is motivated by another question on math.stackexchange.
From a function $g:X^k\to X$ it is possible to define an iterated function system on $X^k$ with the function $f:X^k\to X^k$ ...

**3**

votes

**0**answers

76 views

### Question about a length inequality in algebraic dynamics

Let $X$ be a Noetherian scheme. Let $f\colon X\rightarrow X$ be an integral self-morphism. If $x\in X$ is a closed point, I will write $\mathcal{F}_{1}^x$ for the coherent sheaf of $\mathcal{O}_X$-...

**3**

votes

**0**answers

392 views

### confusion about rational maps and Fatou components

Dear fellows,
I have come to another conclusion which must be wrong.
Let $f$ be a rational map and let $U$ be a connected but not simply connected open subset of the Fatou set such that $f(U)$ is ...

**3**

votes

**0**answers

248 views

### Picture of the set of discontinuity of degree 2 rational Julia sets

Let $Rat_d$ be the set of all rational fraction of degree $d$ and $X_d \subset Rat_d$ be the bifurcation locus of rational fractions of degree $d$, i.e. the closure of the set of discontinuity of the ...