Hierarchies

From Tetration.net

Hierarchies

A very interesting function Abel's equation applies to is $f (z) = e z - 1$ $f(f(z)) = e^{e^z -1} -1$ - generating function for Bell numbers or hierarchies of height 2, a type of non-planar rooted tree. $f(f(f(z))) = e^{e^{e^z -1} -1} -1$ - generating function for hierarchies of height 3. $f n (z)$ - generating function for hierarchies of height n. So a function that is very close to tetration is the generating function a very important class of combinatorial structures - hierarchies of height n.

Hierarchies Julia Set

Parabolic Tetration Julia Set

Look at the fractals generated by hierarchies $f (z) = e z - 1$ and parabolic tetration $g(z) = {}^z (e^{1/e}) \approx {}^z 1.444$ . They produce ALMOST the same fractal. That doesn't mean that $f n (z) = g n (z)$ , but it does mean that there is an $h (z)$ such that $f n (z) = h - 1 (g n (h (z)))$ . Functions $f (z)$ and $g (z)$ are topologically conjugate. If $h (z)$ was known then a linearization of either $f (z)$ or $g (z)$ would immeadiately give the linearization of the other function.