Questions tagged [complex-dynamics]
Dynamics of holomorphic transformations; Mandelbrot and Julia sets.
204 questions
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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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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 ...
- 377
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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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3
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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 ...
- 3,022
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3
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Computing the maximum modulus
For each $a\in \mathbb C$ define $f_a:\mathbb C\to \mathbb C$ by $f_a(z)=\exp(z)+a$. I am primarily interested in real values $a\in (-\infty,-1)$.
For each $r\in [0,\infty)$ define $M_a(r)=\max\{|f_a(...
- 3,317
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1
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Symmetries for Julia sets of perturbations of polynomial maps
This is a naive question. Consider the
Julia sets
of the map
$$ z \mapsto z^n + \lambda / z^k $$
with $z,\lambda \in \mathbb{C}$, and the exponents $n,k \in \mathbb{N}$.
For example, for $n=k=3$, ...
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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 ...
- 1,706
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Analytic ODE with complex time
Suppose we have a complex vector field on $\mathbb{C}^n$ which is analytic and has $|DV| < L$ on ball $B_r$ with radius r.
I would like to understand:
1) if there exists an analytic flow $\phi_t(x)...
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2
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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 $\beta$ ...
- 143
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1
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Attractive Basins and Loops in Julia Sets
I recently learned about the Mandelbrot set for the first time from a presentation by some undergraduates in honor of Mandelbrot's death. The presentation was short and by non-experts so I left with ...
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1
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Can doubly parabolic Blaschke product (BP) contained in another doubly parabolic BP?
Let $f:\mathbb{D}\rightarrow\mathbb{D}$ be a degree $d$ doubly parabolic Blaschke product with Denjoy-Wolff point at $z=1$. That is, $f(1) = 1$, $f'(1)=1$ and $f''(1)=0$.
Let $U \subset \mathbb{D}$ be ...
3
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1
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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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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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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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1
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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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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 $f:X^k\to X^k$ ...
- 709
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80
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Confusion on the assumption when discussing the kneading invariants for unimodal maps
A unimodal map is a continuous map $f:[0,1]\longrightarrow [0,1]$ such that there is only one turning point (critical point), denoted by $c$, and $f(0)=f(1)=0$.
Unimodal map is related to kneading ...
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Does there exist a Runge Fatou-Bieberbach in each Fatou-Bieberbach domain?
A Fatou-Bieberbach domain $\Omega \subseteq \mathbb{C}^n$ is a domain that is a proper subset of $\mathbb{C}^n$ and is biholomorphic to $\mathbb{C}^n$. A domain is said to be Runge if for each ...
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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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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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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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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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3
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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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4
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347
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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 ...
- 1,706
2
votes
2
answers
2k
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Composition of circle inversions
I would like to understand the map of $\mathbb{C}$ to $\mathbb{C}$
that results by iterating inversion in a unit circle.
Let $f(z)$ for $z \in \mathbb{C}$ invert $z$ in a unit circle
centered on $q_1$,...
- 151k
2
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1
answer
187
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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 ...
- 45
2
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3
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436
views
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 ...
- 7,722
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1
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Dynamics of $e^z+z^2+z+1$
Question: How to find the dynamics of $f(z)=e^z+z^2+z+1$ also how to find the escaping set ?
Since critical points control the dynamics, I want to find the critical points first, but I am not getting ...
2
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1
answer
148
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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 ...
2
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1
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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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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
2
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1
answer
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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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1
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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 ...
- 388
2
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2
answers
819
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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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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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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 (...
- 3,317
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1
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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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1
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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 ...
user78249
2
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1
answer
251
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Help determining the asymptotic behavior of an integral involving rational functions.
Let $\phi:\mathbb{P}^1\to\mathbb{P}^1$ be a rational function of degree $d\geq2$. How can one prove, using the normalized spherical measure, that
$$\int_{\mathbb{P}^1(\mathbb{C})}|(\phi^n)'(z)|\ d\mu (...
user4324
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Conjugacy between piecewise linear circle maps
Let $\mathcal{M}$ the Mandelbrot set,
$\mathcal{M}=\{c \in \mathbb{C}: \{Q_c^n(0) \}_{n \in \mathbb{N}} \text{ is bounded, where } Q_c(z)=z^2+c \}$
And let the hyperbolic or stable component, $H_n=\{ ...
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Finding a branch cut or a branch point [closed]
Is there a way to find a branch cut or a branch point, through which a curve over a complex function goes, or in general in some region of complex function, say $\ln(f(z))$, using numerical methods or ...
- 67
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Understanding a more intricate Schwarz reflection principle--A question about Tetration
everyone. This is going to be a long question as it requires a good amount of back story in theory. This question is mostly along the lines: "I think this should happen, and I think my proof is ...
- 388
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When is replacing the prefix of an angled internal address a valid operation?
While working on an artwork exploring patterns in the Mandelbrot set fractal, I constructed an angled internal address by:
$$
1 \overset{1/2}\longrightarrow 2 \overset{1/2}\longrightarrow 3 \overset{1/...
- 111
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55
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Integral curves of rational vector fields and approximations
The following is the formal statement of a conjecture that feels almost obvious, but I cannot find a reference for it. The idea is that one can obtain the integral curves of a vector field $V(z)$ by ...
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How to compute expansion factors for hyperbolic rational maps?
It is a commonly-referenced result about certain rational maps acting on $\mathbb{\hat{C}}$ that they are expanding on a neighborhood of their Julia sets. A sufficient condition to be expanding is ...
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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. ...
- 35
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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 ...
- 2,106
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1
answer
258
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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 ...
- 1,706
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1
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200
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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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vote
1
answer
476
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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 ...
- 356