All Questions
Tagged with complex-dynamics reference-request
29 questions
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 $q_1$,...
- 4,713
10
votes
1
answer
705
views
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 ...
- 91.8k
1
vote
0
answers
61
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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 ...
- 3,599
6
votes
2
answers
249
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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, ...
- 6,548
1
vote
1
answer
486
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Mandelbrot set and logistic map connection
I'm currently writing an undergraduate thesis on chaos theory with a particular focus on the connection between the Mandelbrot set and the logistic map. I have found scattered posts on this site, ...
- 91.8k
7
votes
1
answer
248
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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, ...
- 63.9k
5
votes
0
answers
103
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wild julia sets
Using the Baire category theorem, we may show that most simple closed curves satisfy the following property: any segment between an interior point and an exterior point of the curve intersects the ...
- 18.7k
5
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2
answers
1k
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Who proved that the Mandelbrot set's Julia sets are locally connected?
I'd be greatly interested in a reference to the respective article.
Was it Douady? Julia? Hubbard? Fatou?
Bonus question: Who gave the proof that can be found in the Orsay notes?
EDIT: The question ...
- 91.8k
1
vote
0
answers
61
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Modulus estimate with intersecting annuli (quasi-additivity)
In general for annulus $A\subset \mathbb{C}$ if $A_{1},A_{2}....\subset A$ are disjoint annuli inside it, then we have
$$mod(A)=\frac{1}{2\pi}\int_{A}\int_{A} \frac{1}{|z|^{2}}dz>\frac{1}{2\pi}\...
- 5,474
15
votes
3
answers
2k
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Convergence of Newton's method
For a polynomial $P$ of degree $n$ with real coefficients and with $n$ distinct real roots, the Newton's method $z_{n+1} = z_n - {P(z_n) \over P'(z_n)}$ converges for almost all initial values $z_0$ ...
- 18.7k
2
votes
1
answer
75
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Are the immediate basin of these exponential maps simply connected?
This is a simple question that I direfully need an answer for. If the response is in the negative, I can work with it. If the response is in the positive, I can also work with it. I just can't seem to ...
- 91.8k
1
vote
1
answer
104
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Reference Request: Siegel Center Problem
Does anyone have a reference to where I may find a statement of the problem and perhaps (but not required) some elementary dicussion of Siegel's Center Problem?
- 16.5k
4
votes
1
answer
116
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Can iterates of a non-polynomial function be bounded by an exponential indefinitely?
Assume $f$ is an entire non-polynomial function of arbitrarily small exponential order ('zero'th order' if you're into calling it that). Is it possible that for all $n$ we have
$$|f^{\circ n}(z)| <...
- 7,239
6
votes
0
answers
223
views
Reference request: Complex geodesic flow
Can someone suggest a book on complex geodesic flow? I am interested in it mainly because I was told these form a very useful class of Riemann surface laminations. Of special interest to me is the ...
5
votes
1
answer
353
views
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} &...
CommunityBot
- 1
1
vote
0
answers
294
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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 $\mathbb{C}$
$$\...
CommunityBot
- 1
1
vote
1
answer
379
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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}^\...
- 63.9k
1
vote
0
answers
52
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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 ...
CommunityBot
- 1
4
votes
1
answer
206
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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 ...
- 445
12
votes
4
answers
988
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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, ...
- 445
1
vote
0
answers
60
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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 ...
- 381
8
votes
3
answers
2k
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Harmonic level sets and boundary data
This is probably a classic problem, so a good reference book or paper to get me started on this type of question would be great:
Let $\mathbb{D} \subset \mathbb{C}$ be the unit disk with boundary $\...
CommunityBot
- 1
3
votes
1
answer
166
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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
- 1
3
votes
3
answers
1k
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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 ...
- 3,709
24
votes
1
answer
2k
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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 ...
- 91.8k
21
votes
1
answer
917
views
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^...
- 6,548
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 ...
CommunityBot
- 1
3
votes
1
answer
190
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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 $f:X^k\to X^k$ ...
- 60.5k
3
votes
0
answers
88
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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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