Oh yes, I am well aware of that. Never heard it called a Devaney hair before though, lol.
I think I was misunderstanding something. I was referring to the fact that iterated log's have attractive cycles at these points. So if you draw a spline in a certain manner; and expect the iterated logarithms to be stable on these splines; then it cannot be Kneser's Tetration. Because Kneser's tetration tends to L,L* as we iterate the logarithm; and doesn't cluster towards any periodic points. Which, I wasn't so sure, but it seems like this idea may force the iterated log to be normal on the spline (given the correct key). I think I may be missing something though... I guess it could still produce Kneser's solution; so long as it's not normal for whatever Kneser's key is in a neighborhood of that point. Where I'm referring to kneser's key as the key such that,
\(
\log_{s_k} \text{tet}_K(s_k) = \text{tet}_K(s_k -1)\\
\)
And after writing this... I think I see what my error was; I was assuming the algorithm used to draw the spline was in itself a pull back using logarithms. But that's not the case; you're just using the pull back to find the periodic points; and then constructing the spline using fatou.gp (or your polynomial method). That was a silly mistake of me. For some reason I thought you were somehow drawing the spline using the pull back; and that it looked like it would make the iterated log converge about that periodic point. But you're not doing that at all.
That was silly of me. Now it makes more sense, a lot more sense, what you're trying to do. Lol, I get it now. You can ignore my comments, I was misunderstanding something greatly! lolll
Regards, James
I think I was misunderstanding something. I was referring to the fact that iterated log's have attractive cycles at these points. So if you draw a spline in a certain manner; and expect the iterated logarithms to be stable on these splines; then it cannot be Kneser's Tetration. Because Kneser's tetration tends to L,L* as we iterate the logarithm; and doesn't cluster towards any periodic points. Which, I wasn't so sure, but it seems like this idea may force the iterated log to be normal on the spline (given the correct key). I think I may be missing something though... I guess it could still produce Kneser's solution; so long as it's not normal for whatever Kneser's key is in a neighborhood of that point. Where I'm referring to kneser's key as the key such that,
\(
\log_{s_k} \text{tet}_K(s_k) = \text{tet}_K(s_k -1)\\
\)
And after writing this... I think I see what my error was; I was assuming the algorithm used to draw the spline was in itself a pull back using logarithms. But that's not the case; you're just using the pull back to find the periodic points; and then constructing the spline using fatou.gp (or your polynomial method). That was a silly mistake of me. For some reason I thought you were somehow drawing the spline using the pull back; and that it looked like it would make the iterated log converge about that periodic point. But you're not doing that at all.
That was silly of me. Now it makes more sense, a lot more sense, what you're trying to do. Lol, I get it now. You can ignore my comments, I was misunderstanding something greatly! lolll
Regards, James
