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Dynamics of holomorphic transformations; Mandelbrot and Julia sets.

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55 views

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 $y \in F$, $\...
4
votes
2answers
207 views

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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3answers
848 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 $z_0$ ...
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0answers
55 views

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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0answers
86 views

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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1answer
41 views

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 ...
3
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1answer
68 views

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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1answer
73 views

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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1answer
74 views

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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1answer
104 views

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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1answer
96 views

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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1answer
68 views

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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1answer
117 views

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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votes
1answer
152 views

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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0answers
179 views

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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0answers
100 views

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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2answers
184 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\...
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votes
0answers
165 views

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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0answers
134 views

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 $\mathbb{C}$ $$\...
6
votes
1answer
167 views

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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2answers
182 views

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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0answers
139 views

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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0answers
46 views

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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1answer
355 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. ...
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1answer
148 views

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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votes
1answer
144 views

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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0answers
78 views

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 ...
2
votes
1answer
62 views

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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1answer
847 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 ...
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0answers
85 views

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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3answers
333 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 ...
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5answers
3k views

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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4answers
7k views

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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votes
2answers
204 views

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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0answers
95 views

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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votes
1answer
92 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
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1answer
243 views

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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0answers
633 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 ...
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0answers
46 views

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 ...
12
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0answers
238 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 ...
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0answers
160 views

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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votes
0answers
180 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 ...
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votes
2answers
438 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 ...
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3answers
883 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 $M$ ...
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0answers
79 views

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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1answer
141 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 ...
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1answer
108 views

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 ...
2
votes
3answers
348 views

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 ...
2
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4answers
229 views

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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2answers
141 views

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 ...