Pentation fractal

[Daniel]

First off, beautiful fractals as always, Daniel. I wish I understood in greater detail how you constructed them, but I'm a naive programmer. I'm better at moving hex code and numbers around; I suck at describing detailed graphing protocols/systems of construction.

I'd also like to add that this is a very very deep problem.

Even if we just think about Schroder, we have to choose a fixed point, create an iteration, then choose a fixed point of that iteration. Your method isn't exactly Schroder's method, it's closer to the regular iteration method; I have said though, that it's expressible through Schroder. So, the bell polynomial method (your method as I understand it) is similar to Gottfried's; and this generally creates Schroder's method about a fixed point; but a tad different--especially because it handles neutral fixed points.

So, as I see it, you have two paths ahead of you.

Take the regular iteration route, where Kouznetsov develops how to create pentation for pretty much every base. This is very similar to what Bo talked about with his Taylor expansions; but a full on book describing the regular iteration. Which incidentally, is just an alternative language for the bell-polynomial language (for the most part).

Or, if you want to make graphs with \(b \in (1,\eta)\), and all the hyper operators, and for real values, and learn how these iterations would work--keep talking to me.

I assume you chose the first route. I highly suggest Kouznetsov's text book. If you Pm me I'll send you the link to the down load.  He writes about all sorts of iteration theory, and it's done through regular iteration. He has a good long section on pentation which I'm sure you'd love

How about I take both routes. As far as graphing the higher hyper operators go, I'm quite interested.
Define the hyper etas \(\eta_k\) with \(\eta=\eta_2\) as the parabolic case of \(\eta_k \uparrow^k x\) where the multiplier is 1 and described by the Abel functional equation.

I have the following conjectures. \(\lim\limits_{k \to \infty}\eta_k=2\) also that for \(b \in (1,\eta]\) \(j<k \implies b \uparrow^j x > b \uparrow^k x\) I call these "slack" functions because they grown slower without bound.

\(\lim\limits_{k \to \infty}b\uparrow^k x = b\)