Questions tagged [mandelbrot-set]
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12 questions
1 vote
0 answers
73 views
Is there an equivalent to the logistic map for a nonlinear path through some of the other nodules of the Mandelbrot set?
The logistic map can be related to the real axis of the Mandelbrot set, looking at the different cycle lengths as you pass through all the various nodules along the real axis. But there are other ...
2 votes
1 answer
146 views
Mandelbrot boundary and component of $\infty$
Let $M$ be the Mandelbrot set, and $\partial M$ its boundary. So $\partial M$ is the set of those points $z\in M$ such that every neighborhood of $z$ contains a point of $\mathbb R^2\setminus M$. Let $...
- 3,691
11 votes
1 answer
574 views
Is the Mandelbrot set Suslinian?
The Mandelbrot set is known to be (path-)connected and compact. A non-degenerate space with these properties is called a continuum. A continuum $X$ is Suslinian if every collection of non-degenerate ...
- 3,691
19 votes
1 answer
5k views
Is the area of the Mandelbrot set known? [duplicate]
The Mandelbrot set has an area; is it known exactly? If so, how, and what is the value? If not, why is this a hard question to answer?
- 1,010
5 votes
1 answer
373 views
What is the logical complexity of the Mandelbrot Local Connectivity conjecture? (Is it equivalent to a statement of arithmetic?)
Denote by MLC the statement “the Mandelbrot set is locally connected” and MHC the statement “hyperbolic components are dense in the Mandelbrot set” (it is known that MLC implies MHC, and whether ...
- 38k
9 votes
1 answer
859 views
Does the Mandelbrot set have dense interior?
Let $M$ be the Mandelbrot set. Question: Is the interior of $M$ dense in $M$? My intuition is that this is true, and moreover that hyperbolic components of the interior are dense in $M$ as well, and ...
- 2,270
3 votes
0 answers
111 views
How well do Gauss-Legendre quadrature methods fare on "fractal" functions?
The context I'm making your tipical Mandelbrot set viewer, and I have a function $f: ℂ → ℕ$ that counts how many iterations of $$ z_0 = 0 \\ z_{i+1} = z_i^2 + c $$ it takes for a particular point $c$ ...
- 31
19 votes
4 answers
1k views
Combinatorial description of the Mandelbrot set
I have a very naïve question: can one find anywhere a combinatorial description of the Mandelbrot set? Let me try to be a bit more precise: is it possible to encode each of its "bulbs" by ...
- 1,629
2 votes
0 answers
128 views
Is speaking about a fraction of the Mandelbrot's set meaningful?
Sorry if my question is vague, as I have very little background with fractals and measure theory. My question is inspired by a tweet, where a light shone onto the mandelbrot set, and certain rays were ...
- 121
-1 votes
1 answer
252 views
How does a connected Julia set imply a member of the Mandelbrot Set?
I'm doing an introductory online course in complex analysis. In one of the lectures its stated that a complex number $c$ belongs to the Mandelbrot Set iff the Julia set $J(z^2 + c)$ is connected. I ...
- 11
1 vote
1 answer
375 views
Can the Mandelbrot set be designed through inequalities?
Many years ago I found an inequality that directly controlled whether a point $\displaystyle c$ belongs or does not belong to the Mandelbrot set. Roughly, it was something like this: If $\displaystyle ...
9 votes
0 answers
388 views
Discriminants of Gleason's period-$n$ polynomials for the Mandelbrot set
Gleason's polynomials are the sequence of monic integer polynomials defined recursively by $$ \prod_{d \mid n} G_d(c) = (((c^2+c)^2+c)^2+\cdots+c)^2+c \quad \quad \quad [\textrm{$n$ iterates}], $$ for ...
- 13.9k