Convergence
The subject of convergence is the most active research area in tetration and fractional iteration. In a series of threads on
Math Overflow people use the alpha technique, Bell polynomials and matrices to solve for the half iterates of functions. So there are ways to obtain formal power series, but the series must converge in order to be useful. Unfortunately determining if a series converges or not is often challenging.
Let \(f(f(z))=a^z\) then \(f(z)=\sum_k^\infty \frac{1}{k!} c_k z^k\) and \(log(c_k)\) is plotted to give a sense of whether \(f(z)\) is convergent or not. Two caveats here. The first issue is that only the first 300 or 500 terms have been computed, so at best we are looking for indicators for convergence or divergence. The second issue is that resticting the usuage to rational arithmetic seriously limits the number of terms that can be computed. So floating point arithmetic has been used to produce these plots.
From \(1\) to \(e^{1/e} \approx 1.444\)
a=1
Consider \(1^z=1\) so \(1^{1^z}=1^z\) gives \(f(f(z))=f(z)=1^z\). The function \(f(z)=1^z\) is convergent, but only has 1 term, therefore a plot is not included.
a=1.1
This plot appears to be that of a convergent function which isn't suprising since \(a\) is so close to 1.
a=1.2
Around the 260th term, \(f(z)\) changes it's behavior.
a=1.33
At this point, the convergence of \(f(z)\) changes greatly though \(a=1.36\).
a=1.34
Now the dynamics of convergence plots are becoming chaotic.
a=1.35
a=1.3525
This appears to be close to the most extreme chaotic behavior that will be shown.
I was looking for a smooth transition from convergent to divergent behavior, but
found a chaotic transition. I've seen chaos many times at the boundary of two attractors.
In this case I suggest that the two attractors are the convergence of the superattractor at
\(1^z\) and the divergence of \(1.444^z \approx e^{z/e}\).
a=1.355
Note what is being presented here is a description, not a proof. But it is a description made using
considerable computer resources.
a=1.357
a=1.35866
a=1.36
a=1.365
a=1.38
Another principle that appears at play is that there is only so much
total area above and below the curve \(log(c_k)\).
a=1.4
a=1.414
Special Cases
a=1.44467
This is the case for what some call \(\epsilon\), where \(\epsilon=e^{1/e}\approx 1.44467\). It is at \(\epsilon^z\) that tetration has a parabolic rational fixed point.
a=2.716
a=2.87293 + 4.54887 i
Strange attractor with \(a^{a^a} \approx 1 \).
