All Questions
Tagged with complex-dynamics ds.dynamical-systems
88 questions
5
votes
0
answers
216
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 ...
user47305
10
votes
2
answers
647
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 ...
- 609
27
votes
3
answers
948
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 $M$ ...
- 373
0
votes
0
answers
112
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 ...
- 356
1
vote
1
answer
176
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 ...
- 527
3
votes
3
answers
1k
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 ...
1
vote
1
answer
476
views
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
24
votes
1
answer
2k
views
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 ...
- 91.8k
1
vote
1
answer
179
views
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 ...
- 356
12
votes
4
answers
988
views
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, ...
- 3,022
1
vote
0
answers
226
views
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 ...
- 709
3
votes
1
answer
190
views
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
6
votes
2
answers
936
views
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
4
votes
3
answers
1k
views
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 (...
- 445
1
vote
0
answers
149
views
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 ...
- 149
5
votes
1
answer
444
views
Smoothness in Ecalle's method for fractional iterates
Some four years ago I answered my own question on fractional iteration, concluding that there is a half iterate of sine, that is $f(f(x)) = \sin x,$ which is real analytic for $0 < x < \pi$ but ...
- 25.7k
6
votes
0
answers
321
views
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 $\mu$...
- 589
6
votes
3
answers
392
views
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 ...
- 61
6
votes
5
answers
2k
views
Precise location of the Mandelbrot Bulb Attachment to the main Cardioid
Is there an analytical formula for determining the location of the attachment points of the bulbs on the main cardioid? I was told there is an exact parametrization of the boundary of the main ...
- 183
1
vote
1
answer
98
views
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}\...
3
votes
0
answers
88
views
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 $\mathcal{O}_X$-...
- 3,101
4
votes
1
answer
286
views
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
$$
h(f(z))...
- 695
2
votes
3
answers
436
views
Fatou sets and topological entropy
Let us consider a diffeomorphism of a compact real manifold (complex manifold defined over the reals), and let us say that the diffeomorphism is birational. Hence, it extends to a birational map from ...
- 7,722
12
votes
3
answers
387
views
Dynamics in one matrix variable
Are dynamical systems
$$X \mapsto F(X)$$
studied where $X \in \mathrm{M}_n$, $\mathrm{M}_n:=\mathrm{Mat}(n,\mathbb{C})$ or $\mathrm{Mat}(n,\mathbb{R})$, and $F$ is a (properly defined noncommutative)...
- 23.3k
1
vote
2
answers
821
views
complex dynamics in several variables
Dear mathematicians,
I want to know how much advance there has been in complex dynamics of several variables. I am at present reading Carleson's book on Complex Dynamics on one variables.Curious to ...
- 2,106
4
votes
2
answers
484
views
Algebraicity of the "outer" boundary of the Mandelbrot set
Let $M$ be the Mandelbrot set and let $\lambda\in M, \mu\in \mathbb C$ be algebraic numbers. Let $t_{\lambda,\mu}$ be defined as
$$
t_{\lambda,\mu} = \sup \lbrace t\in \mathbb R\colon \lambda +t\mu \...
- 3,897
17
votes
5
answers
2k
views
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 ...
user39719
9
votes
2
answers
1k
views
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_\...
- 709
32
votes
3
answers
2k
views
How is the Julia set of $fg$ related to the Julia set of $gf$?
Let $f$ and $g$ be complex rational functions (of degree $\geq 2$ if that helps). What can be said about the relationship between $J(fg)$ and $J(gf)$, the Julia sets of the composite functions $f \...
- 27.7k
15
votes
8
answers
4k
views
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 ...
6
votes
4
answers
763
views
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 ...
- 3,101
26
votes
7
answers
2k
views
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 ...
- 3,101
26
votes
6
answers
5k
views
Parametrization of the boundary of the Mandelbrot set
Does anyone know how to parametrize the boundary of the Mandelbrot set? I am not a fractal-geometer or a dynamical systems person. I just have some idle curiosity about this question.
The ...
- 1,277
3
votes
1
answer
776
views
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 ...
- 41
44
votes
3
answers
4k
views
When does iterating $z \mapsto z^2 + c$ have an exact solution?
If one iterates the map $z \mapsto z^2 + c$ there is obviously a simple formula for the sequence one gets if $c=0$. Less obviously, there is also a simple formula when $c = -2$ (use the identity $2 \...
- 20.7k
3
votes
3
answers
865
views
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)...
- 303
10
votes
0
answers
543
views
What is the "category of bifurcations"?
While reading the introduction to this paper by Curtis McMullen, I came to the following (bold added):
In this paper we show that every bifurcation set contains a copy of the boundary of the ...
- 5,992
9
votes
6
answers
2k
views
When does the sequence of iterates of a rational function converge?
Darsh asks at the 20-questions seminar:
Let $f:P^1 \rightarrow P^1$ be rational function.
Can you say when the sequence $\{ f^n(x)\}_n=\{ x,f(x),f(f(x)),\cdots\} $ converges? What about the sequence ...
- 1,059