Questions tagged [complex-dynamics]
Dynamics of holomorphic transformations; Mandelbrot and Julia sets.
204 questions
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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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2
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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 ...
user37691
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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 $\mathbb{C}$
$$\...
user78249
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Are there such things as non-trivial entire semigroups?
I'll state the theorem I am posing up front, and then explain why I think this theorem appears to be true. I am asking if anyone can prove it, or knows references to where it is proved. Please, ...
user78249
2
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2
answers
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Definition of Post-critically finite map
I'm studying the dynamic of the post-critically finite for my master thesis and my professor gave me the problem concerning generalization of post-critically finite ration map. Concretely, let $f : \...
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Is the closed orbit of the Van der Pol equation a stable periodic orbit?
We consider the Van der Pol vector field $$(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 ...
user78249
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1
answer
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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.
...
user78249
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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}^\...
user78249
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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 ...
user78249
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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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1
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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 ...
user78249
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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 ...
user78249
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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 ...
user47305
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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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1
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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 ...
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complex dynamic system and affine algebraic variety
Let $M^n$ be a $n$-dimensional noncompact complex manifold. In "The density property for
complex manifolds and geometric structures II", Dror Varolin showed that some open set of
$M$ is ...
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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^...
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constructing koenigs function
My question is rather simple and I hope someone has some sort of an answer. I am looking for a simple yes or no answer, and a reference if anyone has one.
We have a holomorphic function $f$ defined ...
user78249
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A question on $J(f)$ and $J(f')$
I was confused by the following question for a long time:
Does there exists a transcendental entire function $f$ such that
$J(f)\cap J(f')=\emptyset$ ?
where $J(f)$, ($J(f')$) is the Julia set of $f$...
- 1,706
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Two limit cycles which lie on the same leaf
Edit 1: For a related discussion see this MSE post
I apologize in advance, if this question is obvious:
1)What is an example of a polynomial vector field on $\mathbb{R}^{2}$ with at least two ...
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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 ...
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Stability of singularity in singular holomorphic foliation
For an open subset $U$ of $\mathbb{C}^{2}$ containing $0$ and a holomorphic map $f:U\to \mathbb{C}^{2}$ which has a unique zero at the origin we associate a natural singular holomorphic ...
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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, ...
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What is the state of the art of visualizing bifurcations for "difficult" dynamical systems?
This question is related to my other recent question on MO (although I am not confident that the dynamical system described in that other question is actually "difficult," in the sense that I will ...
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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$ ...
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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 ...
- 1,706
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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} = ...
- 709
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Is there an (almost) dense set of quadratic polynomials which is not in the interior of the Mandelbrot set?
For the parameter plane of complex quadratic polynomials, $(z\mapsto z^2+c)_{c\in\mathbb{C}}$ :
Is it possible to find a part of the parameter plane, scanned with a given limited precision (...
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3
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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 ...
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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 ...
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