Questions tagged [complex-dynamics]
Dynamics of holomorphic transformations; Mandelbrot and Julia sets.
204 questions
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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 $z_0$ ...
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Is a domain of a holomorphic flow pseudoconvex?
Let $Z$ be a holomorphic vector field on $\mathbb{C}^n$. I would like to know whether (it seems that it is) the domain $D_\phi \subset \mathbb{C} \times \mathbb{C}^n$ of a maximal flow $\phi: D_\phi \...
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Is $\partial M_d$ continuously determined by $d$?
This question is inspired by a question on math.stackexchange:
https://math.stackexchange.com/questions/1707291/is-the-generalized-mandelbrot-set-a-fractal-in-the-d-dimension/2575089
The animation ...
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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 \...
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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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Why are the Julia sets so simple? (quadratic family)
I want to know why, when I look at the Julia sets of the quadratic family, I see only a finite number of repeating patterns, rather than a countable infinity of them.
My question is specifically ...
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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 ...
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How many times line segments can intersect a Jordan curve?
I posted a question on math.stackexchange.com but it seems this question might be open
https://math.stackexchange.com/questions/109752/line-segments-intersecting-jordan-curve
Namely,
is there a set ...
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Does the boundary of immediate basin contain a fixed point?
Let $f$ be a rational map of degree $d\geq 2$, and $B$ is a simply connected immediate basin of an supper-attracting fixed point of $f$. I want to know whether there exists a fixed point of $f$ ...
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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?
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Hölder continuity of holomorphic motions
Let $D$ denote the complex unit disk and $X \subset \mathbb{C}$ some subset. Let us consider a holomorphic motion $i \colon D \times X \rightarrow \mathbb{C}$ (denoted $i_\lambda(z)$) meaning for each ...
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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
$$|f^{\circ n}(z)| <...
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Is there some known way to create the Mandelbrot set (the boundary), with an iterated function system?
Is there some known way to create the Mandelbrot set (the boundary),
with an iterated function system (IFS)?
Julia sets can be formed by iterating the two functions $z \mapsto \pm \sqrt{z-c},$
and ...
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Smoothness in Ecalle's method for fractional iterates
Some four years ago I answered my own question on fractional iteration, concluding that there is a half iterate of sine, that is $f(f(x)) = \sin x,$ which is real analytic for $0 < x < \pi$ but ...
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Inverse image of a Jordan curve
If there exists a rational map $R$ from the extended plane $\hat{\mathbb{C}}$ to itself, and a Jordan curve $J$ on the plane, such that $R$ has no critical value on the curve, can we say that the ...
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Finding the "orthogonal" map of a given 1d map
Let $f:\mathbb{C}\to\mathbb{C}$ be meromorphic or even entire. Let $z:\mathbb{C}\to\mathbb{C}$ be such that it holds
$$z(t+1) = f(z(t)).$$
If $f$ is entire and we choose carefully $z$ can be ...
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complex dynamic system and affine algebraic variety
Let $M^n$ be a $n$-dimensional noncompact complex manifold. In "The density property for
complex manifolds and geometric structures II", Dror Varolin showed that some open set of
$M$ is ...
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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 ...
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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 ...
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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 ...
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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 (...
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Uniformization of Riemann surfaces by iso-classical Schottky groups
Let $\Gamma=<g_1, \dots, g_n>$ $\subset PGL_2(\mathbb{C})$ be a Schottky group of rank $n$. The group $\Gamma$ is called classical if there exists a set of $2n$ pairwise disjoint closed balls $\{...
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Fatou Coordinate for function with rationally indifferent fixed point, and repelling fixed point
Lets say I have $f(z)=z^2+c$, with $c=0.35676274578 + 0.32858194507i$. Then $f(z)$ has a fixed point $\kappa_0=0.15450849719 + 0.47552825815i$, which is rationally indifferent with a period $m=5$. ...
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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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Linearizing a power series by conjugation
Let $\mathfrak{I}:=\big\{ \, f:=\sum_{k=0}^\infty f_k z^k \in\mathbb{C}[[z]]\; : \text{s.t. }\; f_0=0 \;\text{ and }\; f_1=1\big\}$. A most basic result about linearization states that, for any $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 ...
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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} &...
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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}$
$$\...
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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}^\...
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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 ...
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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 ...
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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, ...
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Is the area of the Mandelbrot provably computable?
Recall the Mandelbrot set $M$ is the set of points $c$ in the complex plane such that the sequence $z_0 = 0, z_{n+1} = z_n^2 + c$ is bounded. It is well-known that $M$ is a compact set of positive ...
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Effective estimates for circle packing
The Riemann map from a simply connected domain to the unit disc can be approximated by circle packings thanks to a theorem of Rodin and Sullivan. (That is, take smaller and smaller triangulations and ...
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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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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 ...
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A question about Julia set for quadratic family
Let $P_{c}(z)=z^2+c$. It seems from the software that the map between the parameter $c$ and the Julia set $J(P_c)$ is an injective map. Is there some reference about it? Any comments and reference ...
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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 ...
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Summability of iterates of analytic function
This question, although appearing deceptively easy, has resisted many attacks against it. The question, being simple to state, is something rather non-trivial that is rather crucial towards more ...
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Did Gaston Julia ever get to see a computer-generated image of his eponymous set?
I learned from Wikipedia that Gaston Julia died in 1978. Is it known if he ever got to see a computer-generated image of the set named after him?
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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 \...
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Inverse limits of the interval with a single bonding map below the identity
My question is as follows.
QUESTION. Is there a topological description of the class of arc-like continua that arise as inverse limits of $[0,1]$ with a single continuous surjective bonding map $f\...
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Integral Expression in Complex Dynamics
Let $\phi\in \mathbb{C}(z)$ be a degree $d\geq 2$ rational map, which we can write as $\phi = \frac{f}{g}$ for $f,g\in \mathbb{C}[z]$. Let $\omega_{FS}$ denote the Fubini-Study form on $\mathbb{P}^1(\...
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Algebraic proofs of algebraic theorems about algebraically closed fields
It is well-known that the first order theory of algebraically closed fields admits quantifier elimination, whence the theory $ACF_p$ of algebraically closed fields of given characteristic $p$ is ...
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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 ...
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Diophantine approximation in the Julia set
Let $f : \mathbb{CP}^1 \to \mathbb{CP}^1$ be a rational map of degree $q > 1$; or just a quadratic binomial $z^2 + c$, if one prefers. The Julia set $J_f$ is the closure of the repelling periodic ...
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Tying knots in $\mathbb{C}$
As far as I know, there is no deep significance to this question, but I've been playing around with it for a bit and it seems interesting:
Fix a complex number $c$, and consider the map $J_c: \mathbb{...
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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 $\...
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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 ...
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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 ...
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