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Results tagged with fractals
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Fractals deal with special sets that exhibit complicated patterns in every scale. Fractal sets usually have a Hausdorff dimension different from its topological dimension. Examples include Julia sets, the Sierpinski triangle, the Cantor set. Fractals naturally appear in dynamical system, such as iterations in the complex plane, or as strange attractors to continuous dynamical systems, (see Lorentz attractor).
27
votes
Are the algebraic numbers dense everywhere on the boundary of the Mandelbrot set?
Post-critically pre-periodic quadratic polynomials, i.e. those for which the orbit of the critical point $0$ is pre-periodic, are well-known to be dense in the boundary of the Mandelbrot set. (This is …
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23
votes
Accepted
Analysis of the boundary of the Mandelbrot set
Your notation is unusual, and I am not sure whether I entirely understand it.
I shall take your question to mean: Can every point of the Mandelbrot $M$ set be connected to $\infty$ by a path that int …
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14
votes
Parametrization of the boundary of the Mandelbrot set
I am not quite sure what you are asking. The boundary of the Mandelbrot set certainly is not an analytic curve. In fact, a famous result of Shishikura shows that the boundary of the Mandelbrot set has …
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10
votes
Is the area of the Mandelbrot provably computable?
Your statement that the area of the union of bounded hyperbolic components is lower-computable is very plausible.
So if, as conjectured, hyperbolic parameters have full measure, then the area is ind …
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10
votes
Why are the Julia sets so simple? (quadratic family)
As @GNiklasch points out, you seem to be zooming into two places which are both preimages of the same repelling periodic point. So the images of the Julia set are locally related by a conformal map, a …
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9
votes
Accepted
Why are the Julia sets so simple? (quadratic family)
Given that I have started making pictures, I thought it might be worthwhile adding another, shorter, direct answer to your questions, in addition to my longer, more detailed one.
Question 1. Are the …
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8
votes
Accepted
Is there a way to find regions of depth in the Mandelbrot set other than simply poking around?
It is a little bit difficult to answer the question as posed, because there is a question as to what you mean by "depth".
One of the previous answers mentions Misiurewicz points - parameters where th …
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6
votes
Accepted
Is there an (almost) dense set of quadratic polynomials which is not in the interior of the ...
The Hairiness Conjecture, formulated by Milnor and proved by Lyubich ("Feigenbaum-Coullet-Tresser universality and Milnor’s Hairiness Conjecture", Annals of Mathematics, 1999) states that, near any re …
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5
votes
Attractive Basins and Loops in Julia Sets
It seems you are basically interested in an introduction to complex dynamical systems. The books by Beardon, Milnor and Steinmetz all give good introductions.
Regarding your specific questions:
a) T …
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5
votes
Is this a Julia set (and if so, for which function family is it the Julia set)?
The set in question is the bifurcation locus of the family $f_{\lambda}$. It is hence the set of non-normality of the family
$$\bigl(\lambda\mapsto f_{\lambda}^n(\lambda/3)\bigr)_{n\in\mathbb{N}};$$
…
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4
votes
Accepted
Contractibility of connected holomorphic dynamics?
The answer to your question is negative.
EDIT I have added some additional details and made some corrections.
In the entire case, it is possible to construct an entire function with the following p …
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3
votes
Area of the boundary of the Mandelbrot set ?
(Posting my old comment as an answer, as I probably should have done originally.)
Whether the boundary of the Mandelbrot set has positive area is a famous open problem, which has not been solved today …
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1
vote
Accepted
Minimum number of contractions needed to obtain a particular invariant set
An interesting question. Of course there is some ambiguity in the formulation "when can we tell".
Certainly in explicit examples, one may be able to apply ad-hoc methods. For example, things are eas …
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