Dynamics of holomorphic transformations; Mandelbrot and Julia sets.

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### Is the Mandelbrot set weakly self-similar?

A subset $F$ of an Euclidean space $E$ will be called weakly self-similar if for all $x \in F$ there is $\epsilon_x>0$ such that for all positive $\epsilon \le \epsilon_x$ there are $y \in F$, $\...

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### Newton method and Siegel disks

I am looking for a degree 3 polynomial $P$ whose associated Newton's method $z \mapsto z - P(z)/P'(z)$ has a Siegel disk. Is there an explicit example of such polynomial $P$?

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### 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 $z_0$ ...

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### 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 \...

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### Is $\partial M_d$ continuously determined by $d$?

This question is inspired by a question on math.stackexchange:
https://math.stackexchange.com/questions/1707291/is-the-generalized-mandelbrot-set-a-fractal-in-the-d-dimension/2575089
The animation ...

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### Are the immediate basin of these exponential maps simply connected?

This is a simple question that I direfully need an answer for. If the response is in the negative, I can work with it. If the response is in the positive, I can also work with it. I just can't seem to ...

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### Is it known that MLC is sufficient to prove the density of hyperbolic conjecture of rational maps (or not)

Is it known that local connectivity of the Mandelbrot set (MLC) is sufficient prove the density of hyperbolic conjecture of qudratic family.
I wondered is it known that the MLC is not enough (or ...

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### 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 $f$ ...

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### Reference Request: Siegel Center Problem

Does anyone have a reference to where I may find a statement of the problem and perhaps (but not required) some elementary dicussion of Siegel's Center Problem?

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### 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 ...

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### 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)| <...

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### Finding the “orthogonal” map of a given 1d map

Let $f:\mathbb{C}\to\mathbb{C}$ be meromorphic or even entire. Let $z:\mathbb{C}\to\mathbb{C}$ be such that it holds
$$z(t+1) = f(z(t)).$$
If $f$ is entire and we choose carefully $z$ can be ...

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### Is there an example with Area $0<F(f)<\infty$ for some transcendental entire function

It seems that there may be example of a transcendental entire function with finite (but positive) planar area of the Fatou set in Eremenko-Lyubich class. However, I can't not find it in the ...

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### 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 ...

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### Reference request: Complex geodesic flow

Can someone suggest a book on complex geodesic flow? I am interested in it mainly because I was told these form a very useful class of Riemann surface laminations. Of special interest to me is the ...

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### Uniformization of Riemann surfaces by iso-classical Schottky groups

Let $\Gamma=<g_1, \dots, g_n>$ $\subset PGL_2(\mathbb{C})$ be a Schottky group of rank $n$. The group $\Gamma$ is called classical if there exists a set of $2n$ pairwise disjoint closed balls $\{...

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### 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\...

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### Is there a combinatorial analogue of the “sum over all possible paths”?

Addition, multiplication and exponentiation have important combinatorial analogues. By looking at if there a combinatorial analogue of the "sum over all possible paths", I hope to shed light on if ...

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### 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 $\mathbb{C}$
$$\...

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### 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, ...

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### Definition of Post-critically finite map

I'm studying the dynamic of the post-critically finite for my master thesis and my professor gave me the problem concerning generalization of post-critically finite ration map. Concretely, let $f : \...

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### Is the closed orbit of the Vander pol equation a stable periodic orbit?

We consider the Vander 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 ...

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### 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 ...

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### 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.
...

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### 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}^\...

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### 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 ...

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### 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 ...

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### Infinitely renormalizable parameters for quadratic polynomials

Let $P_{c}(z)=z^2 + c$, where $c\in \mathbb{C}$. Did we know that the set of infinitely renormalizable parameters has Lebesgue measure $0$ in complex plane or has Hausdorff dimension $2$?

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### 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 ...

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### Linearizing an operator

This question is more about a curious identity I have come across, than to do with explicit research. The question is somewhat advanced so I'm posting it here rather than on math stackexchange. It ...

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### 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 ...

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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 ...

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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?

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### Summability of iterates of analytic function

This question, although appearing deceptively easy, has resisted many attacks against it. The question, being simple to state, is something rather non-trivial that is rather crucial towards more ...

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### Inverse limits of the interval with a single bonding map below the identity

My question is as follows.
QUESTION. Is there a topological description of the class of arc-like continua that arise as inverse limits of $[0,1]$ with a single continuous surjective bonding map $f\...

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### 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(\...

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### Quantifying the monotonicity property of the hyperbolic metric

Suppose that $X$ and $Y$ are connected hyperbolic Riemann surfaces, with $X\subsetneq Y$. Let $\rho_{X,Y}$ be the density of the hyperbolic metric of $X$ with respect to that of $Y$; then $\rho_{X,Y} &...

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### 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 ...

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### Constructing the external map to a polynomial-like map [closed]

I am reading the paper by Douady and Hubbard, On the Dynamics of Polynomial-Like Mappings, and I am at a loss to understand a crucial step in the construction of the right domain for finding the ...

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### 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 ...

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### Tying knots in $\mathbb{C}$

As far as I know, there is no deep significance to this question, but I've been playing around with it for a bit and it seems interesting:
Fix a complex number $c$, and consider the map $J_c: \mathbb{...

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### 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 ...

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### 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 ...

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### 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 $M$ ...

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### Integrability of the orthogonal complement of a holomorphic vector field on $\mathbb{C}^{2}$

Assume that $$\begin{cases}\dot x=P(x,y)\\\dot y=Q(x,y)\end{cases}$$ is a non vanishing holomorphic vector field on an open subset $U$ of $\mathbb{C}^{2}\simeq \mathbb{R}^{4}$. It defines a two ...

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### 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 ...

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### Holomorphic vector field with infinite separatrix

Let $V=\sum_{i=1}^{n}a_i(z_1,\ldots z_n)\frac{\partial}{\partial z_i}$ be a holomorphic vector field defined on a neighborhood $U\subset \mathbb{C}^n$ of the origin, such that the common zero point of ...

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### reference on complex dynamics

Please someone suggest me some reference on the topic "Complex Dynamics". I want a brief geometric treatment from the root level. I have graduate level background on complex analysis, riemannian ...

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### A question on Ahlfors covering surface

Given a transcendental entire function $f$, and three Jordan domains $D_1$, $D_2$, and $D_3$ such that the closures of the three Jordan domains do not intersect with each other. Then from Ahlfors ...

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### classification of rational map with exactly only one Fatou component

We know that there exists a polynomial the Fatou set $F(P)$ is connected, which is just an attracting basin for infinity.
I have a question: Given a rational function $R$
such that $F(R)$ is ...