Questions tagged [complex-dynamics]
Dynamics of holomorphic transformations; Mandelbrot and Julia sets.
204 questions
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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.
...
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3
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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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1
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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, $J(f_p)$...
CommunityBot
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3
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1
answer
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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 ...
CommunityBot
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1
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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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3
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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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2
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4
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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 ...
CommunityBot
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2
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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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24
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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 ...
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21
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1
answer
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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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1
vote
0
answers
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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 ...
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1
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1
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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 $f$...
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1
vote
1
answer
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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 ...
CommunityBot
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1
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1
answer
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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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1
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0
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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 ...
CommunityBot
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0
answers
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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} = ...
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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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6
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2
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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 ...
- 6,548
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1
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596
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degree of a rational map on infinitely connected fatou component
Given a rational map $f$ on the Riemann sphere, for their Fatou components, we can calculate the relations between the degree $k=\deg(f|_F)$, connectivity number $n=\mathrm{conn}(F)$ and number of ...
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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 ...
- 6,548
10
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1
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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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22
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4
answers
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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 ...
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0
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Periodicities of a Complex Dynamical System
Consider A function $f:\mathbf{C}^2\rightarrow \mathbf{C}$ defined as $$f_{\alpha, \beta}(z,w)=\frac{\alpha}{z}+\frac{\beta}{w}$$ where $\alpha$ and $\beta$ both are complex number.
It is easy to ...
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votes
3
answers
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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-empty ...
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6
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0
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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$...
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3
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392
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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 ...
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1
vote
1
answer
98
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A question for the inverse orbit in the construction of conformal measure
Recently, I read a theorem of existence of conformal measure for the rational map.
I did not understand two places in the proof. The author claims that
there exists an open set $V\subset \hat{C}\...
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1
vote
0
answers
113
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Algebraic set given by sequence of polynomials
When working on some problem, I have end up with a following situation. Suppose
$P(z)=z^d+a_{d-1}z^{d-1}+\ldots+a_1z+a_0$ is a complex polynomial $d\geq2$ and that $\gamma$ and $\delta$ are non-zero ...
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3
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0
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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$-...
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4
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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_{\...
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12
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3
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387
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Dynamics in one matrix variable
Are dynamical systems
$$X \mapsto F(X)$$
studied where $X \in \mathrm{M}_n$, $\mathrm{M}_n:=\mathrm{Mat}(n,\mathbb{C})$ or $\mathrm{Mat}(n,\mathbb{R})$, and $F$ is a (properly defined noncommutative)...
- 23.3k
1
vote
2
answers
158
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general variational principle for the Julia sets of mermorphic function?
I have seen some attempt in considering topological pressure for Julia sets of exponential function, and elliptic function. However, there exists few reference according to my knowledge?
I want to ...
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3
votes
2
answers
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Is the Hausdorff dimension $Dim_{H}(J(f))$ of the Julia set less than 2 for quadratic rational map?
Let $f(z)$ be a quadratic rational map with two Siegel disks which can be normalized to be $$f(z)=z\frac{z+e^{2\pi i\alpha}}{e^{2\pi i\beta}z+1}.$$ If one of the ratation numbers $\alpha$ and $\beta$ ...
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1
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Normal family and arithmetic progression
It is basic fact that for a holomorphic (or mermorphic) map $f$, the family of iterates $\{f^n\}_{n=1}^{\infty}$, is normal if and only if $\{f^{mn}\}_{n=1}^{\infty}$, is normal $\forall m\geq 1$.
...
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2
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3
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Fatou sets and topological entropy
Let us consider a diffeomorphism of a compact real manifold (complex manifold defined over the reals), and let us say that the diffeomorphism is birational. Hence, it extends to a birational map from ...
- 91.8k
3
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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 ...
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0
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Reachability in dynamic random graphs
There has been some advice on how to ask questions. So I reask my question.I have a problem related to graph theory, random graphs or random geometric graphs in special that really confuses me. ...
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1
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2
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complex dynamics in several variables
Dear mathematicians,
I want to know how much advance there has been in complex dynamics of several variables. I am at present reading Carleson's book on Complex Dynamics on one variables.Curious to ...
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4
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2
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484
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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 \...
- 91.8k
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2
answers
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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 ...
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4
votes
2
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677
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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 ...
- 6,548
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1
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If Fatou set has a Multiply connected Fatou component implies every component of F(f) is bounded
Recently when I read a paper about Fatou component, I met the following theorem which cited in Professor
Eremenko's paper "on the iteration of entire functions"
If Fatou set has a Multiply connected ...
- 91.8k
0
votes
0
answers
328
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Limit sets of Fuchsian groups and relation between lifts to $H$ of homotopic maps between hyperbolic Riemann surfaces
Let $f,g : X \to Y$ be homotopic (quasiconformal) maps between hyperbolic Riemann surfaces $X,Y$. Consider their (unique) lifts $\tilde{f},\tilde{g}: H\to H$ , that fix $0,1,\infty $. My question is : ...
- 1,471
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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 ...
- 91.8k
3
votes
0
answers
428
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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 ...
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0
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1
answer
316
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parabolic immediate basins always simply connected?
Edit: So, my original question (stated below) was to find an error in my "proof" that immediate parabolic basins for rational maps are always simply connected.
Since I have not received any answers as ...
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1
answer
359
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Teichmuller Theory question : Beltrami forms on hyperbolic Riemann surfaces whose lifts are smooth upto the boundary of $\mathbb{D}$
Hello, my question is related to Teichmuller Theory. Let $D$ be the open unit disk and $X=D/{\Gamma}$ be a hyperbolic Riemann surface of the Fuchsian group $\Gamma$. In Teichmuller theory, we have ...
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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 ...
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2
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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}. ...
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