Questions tagged [complex-dynamics]
Dynamics of holomorphic transformations; Mandelbrot and Julia sets.
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Accessible points of a simply connected domain
We know that if $U$ is an open subset of $\mathbb{\widehat C}$ (extended complex plane), a point $v\in\partial U$ is called accessible from $U$ if there exists a curve $\gamma:[0,1)\to U$ such that $\...
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Entire function of finite order with deficient value
There are some sufficient conditions for an entire function to have a finite deficient value e.g., if the order $\rho$ of an entire function $f$ is such that $2<\rho<+\infty$ with all but ...
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Multiply connected Fatou component of an entire function
This question may be trivial but still I want to know the answer.
Question: Is there any necessary condition (except boundedness of the Fatou component) for the existence of a multiply connected Fatou ...
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Is the Mandelbrot set Suslinian?
The Mandelbrot set is known to be (path-)connected and compact. A non-degenerate space with these properties is called a continuum.
A continuum $X$ is Suslinian if every collection of non-degenerate ...
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The number of components of the preimage of a continuum for a polynomial
Given a polynomial f with degree d, we can define a dynamical system $(\mathbb{C}, f)$. If we have a proper continuum $N \subset \mathbb{C}$, it is known that the set $f^{-1}(N)$ has at most $d$ ...
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Any theorem shows that flowmap $\phi_{\sum_{i=1}^n a_i f_i(x)}^\tau$ can be approximated by $\phi_{f_{\theta(t)}(x)}^{\tau'}$?
Given a control family $F:=\{f_1,\dotsc,f_n\}$, and $\phi_f^\tau(x)$ is the flowmap of the dynamical system
$$
\begin{cases}
z'(t)=f(z),\\
z(0)=x,
\end{cases}
$$ at end time point $\tau$.
Suppose $a_i&...
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Orbit closure of two elliptic Möbius transformations
Let $g_1$ and $g_2$ be two elliptic Möbius transformations of infinite order in $\mathrm{Aut}(\bar{\mathbb{C}})$. If $\mathrm{Fix} (g_1) \cap \mathrm{Fix} (g_2) = \emptyset$, then can we deduce that $...
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Is the area of the Mandelbrot set known? [duplicate]
The Mandelbrot set has an area; is it known exactly? If so, how, and what is the value? If not, why is this a hard question to answer?
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$f(f(z)) = z , f(\exp(z)) = \exp(f(z)) $?
While talking about tetration with my friend the following idea (re)occured.
$$f(f(z)) = z ,\quad f(\exp(z)) = \exp(f(z)) \tag{A}\label{A}$$
or variations of it like the weaker
$$f(f(f(f(z)))) = z ,\...
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Is the closed orbit of the Van der Pol equation a stable periodic orbit?
We consider the Van der Pol vector field $$(1) \;\;\;\;\;\; \begin{cases} x'=y-(x^3-x)\\ y'=-x\end{cases}$$ on $\mathbb{R}^2.$
It is well known that this equation has a unique limit ...
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Uniformization of Julia sets and lacunary series
If the complement of a Julia set of quadratic polynomial z^2+c is locally connected and simply connected, it is uniformized by the complement of the unit disk. Consider the uniformization map and its ...
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Dense orbits for a rational map
Given a complex rational function $f$ and $z\in \mathbb C$, let $O^+(z)=\{f^n(z):n\geq 1\}$. Let $$D(f)=\big\{z\in \mathbb C:\overline{O^+(z)}=J(f)\big\}.$$
So $D(f)$ is the set of points whose (...
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Jordan curve boundaries of Fatou components
Let $f:\mathbb C\to \mathbb C$ be a rational map and let $J(f)$ and $F(f)$ denote the Julia and Fatou sets of $f$, respectively.
Let $\mathcal S$ be the set of all boundaries of Fatou components. ...
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Finding a branch cut or a branch point [closed]
Is there a way to find a branch cut or a branch point, through which a curve over a complex function goes, or in general in some region of complex function, say $\ln(f(z))$, using numerical methods or ...
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What about the other $f$ such that $f(f(x)) = \sin(x)$?
This question is inspired by the big MO question here; and also inspired by the big MO question here. The premise of this question requires a backdrop on fractional iteration; so I'll start slow.
...
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A follow up question related to entropy
For a self-map $\varphi:X\longrightarrow X$ of a space $X$, many important notions of entropy are defined through a limit of the form $$\lim_{n\rightarrow\infty}\frac{1}{n}\log a_n,$$ where in each ...
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How to find the hyperbolic dimension of map $f(z) = z^2$ of $\overline{\mathbb{C}}$ onto itself?
I am reading the research article "The Hausdorff dimension of the boundary of Mandelbrot set and Julia sets" by Shishikura. In his article, he defined hyperbolic sets and hyperbolic ...
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Persistence of irrationally indifferent periodic points
I am trying to see when an irrationally indifferent periodic point persists for a holomorphic family of rational maps on the Riemann sphere. This is a question I am currently pondering after reading ...
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What is the logical complexity of the Mandelbrot Local Connectivity conjecture? (Is it equivalent to a statement of arithmetic?)
Denote by MLC the statement “the Mandelbrot set is locally connected” and MHC the statement “hyperbolic components are dense in the Mandelbrot set” (it is known that MLC implies MHC, and whether ...
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Examples of hyperbolic set and J-stable sets
I am reading the research article "The Hausdorff dimension of the boundary of Mandelbrot set and Julia sets" by Shishikura. The following two definitions are given without any examples in ...
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Does the Mandelbrot set have dense interior?
Let $M$ be the Mandelbrot set.
Question: Is the interior of $M$ dense in $M$?
My intuition is that this is true, and moreover that hyperbolic components of the interior are dense in $M$ as well, and ...
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Power series expansion of the Koenigs function
Given a non-zero holomorphic function $f$ fixing $0$ which isn't a Mobius transform, the Koenigs function of $f$, which we'll call $h$, is the function which linearizes $f$ in the sense that
$$
h(f(z))...
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If I have zeros at the vertices of an icosahedron, where should the poles go?
I've been tinkering with Newton's method applied to polynomials. E.g., Newton's method for $z^5 - 1 = 0$ gives:
There aren't a lot of symmetric patterns of finite sets of points in the plane, so I ...
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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 ...
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Borel summation and the Abel function of $e^z-1$
This is a question that has bothered myself and Gottfried Helms a fair amount of late. He has made his case for the following result, but a proof escapes both of us. The question is deceptively simple,...
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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 ...
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Finding the repelling fixed point of an exponential, knowing only its attracting one
This question has been bugging me for a while, I have an answer that is working sufficiently for the program I'm using, but it is a tad slow, and let's say imprecise. It is not an overtly difficult ...
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On entire functions with polynomial Schwarzian derivative
The Schwarzian derivative of an entire holomorphic function $f$ is defined as
$$Sf:=\left(\frac{f^{''}}{f'}\right)'-\frac{1}{2}\left(\frac{f^{''}}{f'}\right)^2.$$
In the following, we only consider ...
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Positive integration on P^1
Let $u: \mathbb{P}^1(\mathbb{C}) \longrightarrow \mathbb{R}$ be a smooth function s.t. $u$ is invariant under complex conjugation and $\displaystyle \int_{\mathbb{P}^1(\mathbb{C})}u \; \omega_{\mathrm{...
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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_\...
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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 ...
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A curious observation on the elliptic curve $y^2=x^3+1$
Here is a calculation regarding the $2$-torsion points of the elliptic curve $y^2=x^3+1$ which looks really miraculous to me (the motivation comes at the end).
Take a point of $y^2=x^3+1$ and ...
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A question about decompositions of rational functions
Let $f_1,g_1,f_2,g_2$ be non-constant rational functions on the Riemann sphere (i.e. elements of $\Bbb{C}(z)-\Bbb{C}$) satisfying $f_1\circ g_1=f_2\circ g_2$. Suppose there is a prime number $p$ such ...
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Tiling the plane with finitely many congruent pieces
Suppose $A_1,\dots,A_n$ are measurable subsets of the plane that are all related by rigid motions such that $|(A_1 \cup \dots \cup A_n)^c| = 0$ and $|A_i \cap A_j| = 0$ for all $1 \leq i < j \leq n$...
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Ahlfors' proof of Bloch's theorem
In his pioneering paper An extension of Schwarz's lemma, Ahlfors proves the lower bound on the Bloch constant $B \geq \frac{\sqrt{3}}{4}$. The proof of this lower bound proceeds as follows:
Let $W$ be ...
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Understanding a more intricate Schwarz reflection principle--A question about Tetration
everyone. This is going to be a long question as it requires a good amount of back story in theory. This question is mostly along the lines: "I think this should happen, and I think my proof is ...
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the (non-existent) group of conformal transformations
In physics intros to 2d conformal field theory, people often talk about the "group of conformal transformations". Of course, that's not a group but rather a pseudo-group... that's not what ...
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Holomorphic dynamical systems defined on a contractible bounded open subset of $\Bbb{C}^n$
Let $U$ be a contractible bounded open subset of $\Bbb{C}$. There is a standard classification of possible dynamical behaviors of holomorphic maps $f:U\rightarrow U$:
Attracting Case: There is an ...
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Renormalization in physics vs. dynamical systems
I am studying complex dynamics, so to me renormalization of a dynamical system means something like a rescaled first-return map on (a subset of) the underlying space. I understand that in quantum ...
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When is replacing the prefix of an angled internal address a valid operation?
While working on an artwork exploring patterns in the Mandelbrot set fractal, I constructed an angled internal address by:
$$
1 \overset{1/2}\longrightarrow 2 \overset{1/2}\longrightarrow 3 \overset{1/...
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Is there a way to find regions of depth in the Mandelbrot set other than simply poking around?
Ie, is there a way to probe it for regions of depth that involves a function, the domain of which is the Mandelbrot set itself, or a part of that set?
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On the relevance of the property $\exp^{\circ a}(\exp^{\circ b}(z))=\exp^{\circ a+b}(z)$ for the *fractional* iteration ("tetration")
In the concept of fractional iteration of the exponential function ("tetration") the property of $$\exp^{\circ a}(\exp^{\circ b}(z))=\exp^{\circ b}(\exp^{\circ a}(z))=\exp^{\circ a+b}(z) \...
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Formal group law and Koenigs function conjecture?
Let $f(x,y)$ be a symmetric real function and a formal group law
$$G(x + y) = f(G(x),G(y)). \tag{1}$$
Consider the equation
$$ h(2x) = f(h(x),h(x)) = A(h(x)). \tag{2}$$
This equation has many ...
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Hausdorff dimension of Julia set
Can anyone show me the proof "Hausdorff dimension of Julia set is strictly positive"?
For purpose to prove this we might have to prove the green function of basin of attraction to infinity ...
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Lower semicontinuity of the number of attracting periodic points of a holomorphic family of rational maps?
Recently I have been reading the book Mathematical Tools for One-Dimensional Dynamics.
In the proof of the theorem 5.4.2, authors use the following fact that the number of attracting periodic points ...
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Exponential iterates of a complex number
Let $f:\mathbb C\to \mathbb C$ be defined by $f(z)=e^z-1$. Let $f^n$ denote the $n$-fold composition of $f$.
In my new paper Erdős space in Julia sets I show that $$Z:=\{z\in \mathbb C:\lvert\...
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Julia set containing smooth curve
I have two realted questions.
Let $R$ be a rational function on $\mathbb{C}$ with degree at least 2. We denote by $\mu$ the measure of maximal entropy for $R$ and recall that the Julia set coincides ...
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Cutting a Julia set into infinitely many pieces at finitely many points
Let $f\colon \widehat{\mathbb{C}}\to \widehat{\mathbb{C}}$ be a rational function of degree two or greater whose Julia set $J_f$ is connected. If $S\subseteq J_f$ is a finite set of periodic points, ...
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Clarification about the process of naturally endowing a space with a Riemann orbifold structure supported on a sphere
I am having some difficulties understanding an argument in a proof. Here is an excerpt from Lyubich–Peters - Classification of invariant Fatou components for dissipative Henon maps, first geometric ...
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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 the field of complex dynamical systems. According to my knowledge, my friends told me that there are many ...
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