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1h
revised Prove that the Mandelbrot Set Is A Closed Set
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1h
comment Question About Filled Julia and Julia Sets
@Overachiever I completed the argument more explicitly, rather than just providing hints. To address your specific questions, it seems that you have one point of fundamental confusion. Your original question does not concern the boundary of the unit disk. That's just a specific case, namely when $c=0$; most Julia sets are much more complicated than this. So there's no reason to suppose that $|Q_c(z_0)|=1$ when $Q_c(z_0)\in J_c$. Rather, you should draw conclusions on points in a neighborhood of $z_0$ from assumptions involving corresponding points in a neighborhood of $Q_c(z_0)$.
1h
revised Question About Filled Julia and Julia Sets
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11h
answered Question About Filled Julia and Julia Sets
Aug
1
comment Is the function $f(x) = |x|$ convex?
There was a very similar discussion in this discussion.
Aug
1
answered Advantages to learning Sage?
Jul
31
comment If the Chaos Game result is a Sierpinski attractor when the random seed is a sequence (Möbius function), does it imply that the sequence is random?
Why don't you use the ternary digits of $\pi$ or some other interesting number to determine your sequence. Or you might try a sequence generated by an interesting cellular automaton or a gray code or the iteration of a chaotic function. These are all deterministic but likely to generate the full Sierpinski triangle.
Jul
31
comment If the Chaos Game result is a Sierpinski attractor when the random seed is a sequence (Möbius function), does it imply that the sequence is random?
Neat construction! However, I don't think that the set of limit points of the sequence being the same as the Sierpinski triangle implies that the process generates a nice image of the Sierpinski triangle. It's quite likely that points will not be uniformly distributed over the attractor. In technical terms, the self-similar measure generated by your procedure is very different from that generated by a random procedure. It might very well be concentrated on a line segment and that's image the procedure would generate.
Jul
30
comment Are there any other non-differentiable that came be constructed from summation besides the Weierstrass function?
If $g(n,t)$ are the terms in any sequence that sum to the zero function, then you can add them to the $f$s and maintain the sum. Certainly, you can choose the $g$s to destroy any periodicity that might have been present in the original terms.
Jul
23
revised Wolfram Alpha formula not working
added 3 characters in body
Jul
23
comment Wolfram Alpha formula not working
@GregVoit The question is not at all about Mathematica; it's about WolframAlpha which is quite different and definitely not a Mathematica interpreter.
Jul
22
comment Box-Counting Dimension with finite resolution
@Bob Well, you lost me there. I'd say an $n$-dimensional grid looks, well, $n$-dimensional regardless of the resolution. Now, if you have an image that consists of a large number of line segments, then it might appear to have a fractal dimension at large resolution while, at small scales, the one-dimensional behavior of those segments might dominate. Is that the point you're trying to make? Regardless, we certainly don't expect $\log(N_{\varepsilon})/\log(1/\varepsilon)$ to be constant. If it's close to constant over a wide range, then that constant could be called the fractal dimension.
Jul
22
comment Box-Counting Dimension with finite resolution
@Bob Right - I'm not sure I see much difference between your statement and mine. I guess the main point that I should really emphasize is that the relationship should hold over many orders of magnitude. Thus, we measure $\log(N_{a^k})$ for some constant $a$ larger than $0$ and (much) less than $1$ as well as for a large number of $k$s. If the relationship between $\log(N_{a^k})$ and $a^k$ holds up over that range, then we see that the fractal object behaves the same on a wide range of scales. This is somewhat like self-similarity.
Jul
22
answered Box-Counting Dimension with finite resolution
Jul
22
comment Example of polynomial in dynamics
@JimBelk Have you noticed this in V10: JuliaSetPlot[2 z^3 - 3 z^2 + 1/2, z]?
Jul
21
answered Construction of Rauzy Fractals with substitutions without a fixed point
Jul
20
comment Example of polynomial in dynamics
You can rule out the mulitbrot family, since they all have a single critical point, namely the origin.
Jul
20
comment Example of polynomial in dynamics
Very nice! It might be worth pointing out that it's easy to show that $1$ is in the Julia set using the characterization that the Julia set is the closure of the repelling fixed points. First, $-1/2$ is in the Julia set since it's a repulsive fixed point (i.e. a point of period $1$). Then, $1$ is in the Julia set since it's the inverse image of $-1/2$ and the Julia set is backward invariant.
Jul
15
comment Why do we say “radius” of convergence?
@ElliotG Your question in the title is 'Why do we say “radius” of convergence?' This response is an important part of that answer.
Jul
12
reviewed Approve Topological Semi conjugacy between Henon map and Logistic Map