Base -1

[sheldonison]

Tetration base 1 is a constant function equal to 1, for x>-1, and has a discontinuity at -1. Or at least that's the solution if \( \\[15pt]

{^{x}1=\lim_{b \to 1} ^xb} \)... I wonder if there is a family of continuous solutions, and this one is the evolvent.

b=-1 has two primary fixed points. both repelling; I'm not sure what the other fixed points are.
-1,
0.266036599292773 + 0.294290021873387*I

Pairs of repelling fixed points can sometimes be used to build analytic complex valued solutions, with the property that tet(-1)=0. As you discovered, there also appears to be another family of solutions for 0<b<1, which is a damped oscillator type solution. So it quickly appears that there is no uniqueness what so ever... with an infinite number of interesting solutions possible.

I find it easier to work with the conjugate base, and analyze iterating the function:

\( y \mapsto \exp(y)-1 + k\;\; \) instead of \( z \mapsto b^z \) where

\( k = \ln(\ln(b))+1\;\; \) and \( z = (y-k+1)\cdot \exp(1-k)\;\; \) this is a simple linear transformation from y to z

Then analyzing the function for iterating \( z \mapsto b^z \)
is conjugate (or mathematically equivalent) to iterating the function
\( y \mapsto \exp(y)-1+k\;\; \) but this conjugate form is much simpler to work with and understand. k=0 is the parabolic case which corresponds to base \( b=\eta=\exp(1/e)\;\;\; \) k>0 corresponds to Kneser's real valued tetration solution, and \( \;\;k= \pi i + c\;\; \) corresponds to Marraco's bases between 0..1
And the conjugate value of k for b=-1 is \( k=2.14472988584940 + 0.5\pi i \)

I have a series solution for the two fixed primary fixed points; http://math.eretrandre.org/tetrationforu...hp?tid=728 which turns out to have a nice Taylor series solution with \( z=\sqrt{-2 k }\;\; \) and with rational coefficients. I am also in the process of debugging a very powerful generic slog/abel pari-gp program for iterating \( z\mapsto \exp(z)-1+k \) for arbitrary complex values of k. This bipolar Abel function may be unique, based on Henryk's proof, but this solution requires that the Abel function be analytic in a strip between the fixed points. For Marraco's damped oscillating solutions, the Abel function has singularities where the derivative of the sexp'(z)=0. I haven't yet generated any analytic solutions for Marraco's damped oscillating solutions for 0<b<1; that's also a longer term goal.