On the webpage http://tetration.org/Tetration/index.html,
We are supposed to get an explanation of tetration, whatever that means exactly.
In particular I feel the pictures are not well explained.
The first two pictures show “ tetration “ and the last a “ Julia set of 2^z “.
However what is meant by tetration ?? Tetration has many parameters , interpretations solutions etc.
My guess is the pictures are related to z^z^z^... where the base is z and the starting value is z. And then we consider z^^n. And we get limits , double limits , triple limits etc or no ( finite ) limit at all.
And then we color them accordingly.
However , even if that is the case , it should have been stated clearly. It might be something else ?
Also , there are 2 pictures , slightly different. And labeled mysteriously : “ by escape and by period “
What does that mean ??? Did I describe one of them ??
Also there are only finite colors.
And nothing about the pictures is explained , proven or even conjectured.
And the last picture is suppose to be a Julia set of an exponential function.
However , exponential functions are strongly chaotic and arbitrarily close to almost any point is a periodic point.
Hence for most bases I find it hard to interpret a Julia set.
Julia sets are well defined for polynomials and a handful transcendentals.
But without attracting points the Julia set of a transcendental entire function is “ fuzzy “.
Exp(z) has no attracting fixpoints for example and near any point is a periodic point, and the iterations are chaotic.
The pictures appear on other places of the website and are also copied to places like Wikipedia , the tetration forum or archiv. And many more. BUT also without explaining things.
Also I assume the domain is colored and not the range of z^z^...
Also I find the colors not even convincing ? All that connected green space ? All large imaginary are green ? Really ??
Or is that just an illusion and we get more structure when zooming out ?
My friend noticed that the equation
z^z = z does not lie within the shel-tron region ( I presume the red part in pic 2 ) ( the smallest nonreal solution z is not in the red. ) It might be true that every solution to
$$ z^{^n} = z $$
( this is not a power on the LHS but a power tower of size n ) (For all n and z)
Might have its own “ color island “. I mean a one-to-one correspondence here.
Do not get me wrong, I appreciate someone maintaining a site about tetration. But after all these years it should have been made clear imho.
Hopefully you can clarify things.
(* Fractint code*)
TetrationM (XAXIS) {; z = pixel: z = pixel ^ z |z| <= 100000 }
Correct. The Tetration fractal goes back to Lee Skinner's Tetrate algorithm in the Fractint fractal generator. This was much better known twenty years ago. The 'escape' fractal is the exponential map analog to the Mandelbrot set. The 'period' exponential Mandelbrot has a complicated period structure that Fractint also generated.
Perfectly, you described the 'escape fractal'.
Correct again.
The Period fractal is just the inverse of the escape fractal. A point will be black in one fractal and a color in the other. Fractint had a mode to quickly compute the period of forward orbits that didn't escape.
From the comments.
Your concerns about $2^z$ can be addressed by the Julia set of the exponential function $e^z$. The cover of An Introduction to Chaotic Dynamical Systems, by Robert Devaney has a Julia set of the exponential function. See Section 3.8 which is devoted to the topic. All the green area is period three.