Mandelbrot Benoit

[Dallas Querry]

ExampleThere are few really easy examples, because the Julia setis almost always a fractal. When ff has degree00 or 11, things are a bit too easy. The simplest nontrivial exampleis f(z)=z 2f(z) = z^2.

For example, we could ask for which values of cc the set J(f c)J(f_c) is the wholeRiemann sphere. (Answer: none.) But what turns out to be really interestingis to ask for which values of cc the Julia set is connected. In fact:

Trying to digest what the Julia and mandelbrot set is here. Coming from control engineering, it strikes me as a nice way to visualize the difficulties in controlling say a machine, aircraft or powerplant.

Then you have the common engineering task of optimizing the system for something. Assuming that the optimal value is known, you have to navigate your way from where you are in parameter space to the optimal point. You can visualize this as a path.

If you visualize it together with the julia set, your path have to wind its way around not touching the julia set. If the julia set is distressed or disintegrated, that makes for a difficult optimization.

Also, not knowing anything about this area except vague snippets I pick up at odd moments, it almost seems to me that the Mandelbrot set is a kind of moduli space: to each cc there is attached a nice Julia set; the cc here evidently parametrize such nice connected Julia sets according to your account. I was wondering whether a nice moduli space story could be told here. (This may be sort of along the lines of the spectrum story you were hinting at.)

it almost seems to me that the Mandelbrot set is a kind of moduli space: to each cc there is attached a nice Julia set; the cc here evidently parametrize such nice connected Julia sets according to your account. I was wondering whether a nice moduli space story could be told here.

Well, it could be that we need to keep track of more than just the Julia set. What I mean is that the Julia set of a rational function ff is forwards invariant under ff, and therefore comes equipped with an endomorphism.

At one point I was making serious efforts to come up with a good definition of the Euler characteristic of a Julia set, based on its self-similarity. (Since every rational function has a Julia set, this would in turn define a numerical invariant of rational functions.) I made some progress, with help from Mary Rees, but never managed a general definition. I mention this because in that context, it seems to be crucial to take the Julia set along with its endomorphism.

Fine, so let it respect the endomorphism. Why would that prevent us from getting a notion of equivalence of Julia sets? It seems on the contrary: since you are talking about endomorphisms , it seems you have already settled on a notion of morphism between them. So look at the isomorphisms!

The sense of computable here is that you can work out lower bounds on the Hausdorff distance between a given finite set and the desired filled Julia, given oracle access to the parameters. The upshot is that you can write programs to draw most of these filled Julia sets and be confident that they get it right.

On the left-hand side of the analogy, kernels and spectra are very different animals: the kernel of an operator on a vector space VV is a vector subspace of VV, whereas its spectrum is a subset of the ground field.

No, but fortunately we have other methods at our disposal! For instance, any topos has a fundamental pro-group, which for Sh(X)Sh(X) agrees with the usual fundamental group of XX if XX is sufficiently nice (something like locally simply connected, probably). But for other XX, the topos-theoretic fundamental group is usually more informative than the traditional one. This also apparently gets you into shape theory and higher topoi, especially when you try to do higher homotopy theory.

One might imagine doing that by going from the Julia set to an algebraic structure in a two-step process, via the intermediate stage of toposes. However, I have a gut instinct that one might have to be somewhat innovative in the first step. Taking the topos of sheaves might not be the right thing to do. In this context, what seems to matter is not so much functions on open subsets as functions on closed subsets.

The terms in the question are not precisely defined. However, I have a fairly precise idea of what I mean. I gave a system of three equations satisfied by J(f)J(f) and two other spaces called XX and YY. The question is asking: what automorphisms of J(f)J(f) are there, if all we know about J(f)J(f) is that it participates in such a system of equations?

There I drew an example of a simple, non-trivial automorphism of [0,1][0, 1] that can be obtained from this isomorphism alone. (That is, I drew a simple example of a non-identity element of FF.) Here is a simple example of a non-trivial automorphism of J(f)J(f) that can be obtained from this system of equations alone:

I see at the beginning of the Bunge-Lack article that they motivate their constructions by formulating it in terms of intensive quantities on extensive categories of generalized spaces . In particular they show that Topos is such an extensive 2-category of generalized spaces.

I agree. Especially having the relationship between the julia set and mandelbrot set spelled out so clearly.

I revisited the previous discussion of fractals on the n-category cafe and a lot more of the discussion made sense to me now. :-)

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