From complex to real tetration

[Daniel]

Edit: Background
I have finished the first version of my Flow Mathematica function which I have submitted to Wolfram for review. I decompose the problem of tetration and the Ackermann function into that of fractional iteration that my Flow function handles and the classical Ackermann function. Having the flow function gets me 90% of anywhere I want to go working with tetration. 

The following is supposed to be a graph of tetration of e. Hopefully it is a broken implementation of tetration.
The problem is the flow that crosses itself. Being homoclinic is a real problem for defining unique tetration. I wonder if anyone else has ever considered issue of being homoclinic.

\[^xe\]

So instead of dealing with \[^y x\] I can choose to use \[^y (1+x)\]. This is nice because it has a fixed point at x=0 giving \[^y 1\]. This fixed point is also on the real line resulting in a real domain and range.

[bo198214]

I am still not exactly sure what the shown curve is.
I guess it is the image of the function \( ^xe\) of some interval (a,b).
If this is right, which interval is shown?
Also I would assume that the points \(^1e=e\), \(^0e=1\) or \(^{-1}e=0\)  are to be seen somewhere, but I don't see them.

[bo198214]

I am still not exactly sure what the shown curve is.
I guess it is the image of the function \( ^xe\) of some interval (a,b).
If this is right, which interval is shown?
Also I would assume that the points \(^1e=e\), \(^0e=1\) or \(^{-1}e=0\)  are to be seen somewhere, but I don't see them.

Daniel, any clarification?

[Daniel]

I am still not exactly sure what the shown curve is.
I guess it is the image of the function \( ^xe\) of some interval (a,b).
If this is right, which interval is shown?
Also I would assume that the points \(^1e=e\), \(^0e=1\) or \(^{-1}e=0\)  are to be seen somewhere, but I don't see them.

Daniel, any clarification?

The graph is pure bullshit because it is divergent. I took a bit to realize how broken it was. The range is from x=-20 to x=10. Just like logs, I'm having more success with \[^y (1+x)\].