Tetration below 1

[Gottfried]

So let me conclude.

If \( -1<\log(h)<0 \) then we have the negative Eigenvalues \( \log(h)^{2n+1} \) in the power derivation matrix A of \( f(x)=b^x \). Now we compute \( A^t \). It has the Eigenvalues \( (\log(h)^n)^t \). Take for example \( t=\frac{1}{2} \) then we see that \( A^t \) has also non-real Eigenvalues and hence has also non-real entries. Supposed \( f^{\circ \frac{1}{2}} \) had only real coefficients then \( A^{\frac{1}{2} \) would have only real coefficients. So it is clear that \( f^{\circ \frac{1}{2} \) must have some non-real coefficients and is a non-real function.

However perhaps it could be that \( {}^{\frac{1}{2}}b=f^{\circ \frac{1}{2}}(1) \) is real or generally that \( {}^tb=f^{\circ t}(1) \) is real, which I dont believe. Can someone just compute it?

Hmm, I don't know, whether I understand you correctly. If h=1 then all eigenvalues except the first are zero ( = [1,0,0,....]) and the result is always the same, independent of any power of log(h) since the "height" y of the tower occurs only as exponent of the eigenvalues....
Did I misread something obvious?

Gottfried