Universal uniqueness criterion?

[Kouznetsov]

Such a function should have some smooth kink of the phase, in order to avoid the jump; but allow such a jump for the principal branch of its logarithm...
I think about something like \( E(y)=
\Re(L_1) +(1-\Re(L_1))/ \cosh(y) + \Im(L_1) \tanh(y) \)

Can you show me how E(y) would have the behavior as you speculate? Also do the tetrations constructed with other fixed point pairs still satisfy the basic functional equations b[4](z+1) = b^b[4]z and b[4]1=b?

I played it a little bit. We should take an exponential of the expression I wrote and scale the argument; then it reproduces the kink of reasonable size. Of course, such a probe function does not satisfy the tetration equation, but it shows some reasonabe behavior within the strip along the imaginary axis: after the esponentiation, it has just one kink in the upper halfplane and one in the lower halfplane; this is all we need for the function at the first iteration. We may glue it to the asymptotics, and plot the function even before to make the precise evaluation.