Let \( F(x)=b^x \) for some base \( b \).
Then we demand that any tetration \( f(x)={}^x b \) is a solution of the Abel equation
\( f(x+1)=F(f(x)) \) and \( f(1)=b \).
Such a solution \( f \) (even if analytic and strictly increasing) is generally not unique because for example the solution \( g(x):=f(x+\frac{1}{2\pi}\sin(2\pi x)) \) is also an analytic strictly increasing solution, by
\( g(x+1)=f(x+1+\frac{1}{2\pi}\sin(2\pi + 2\pi x))=F(f(x+\frac{1}{2\pi}\sin(2\pi x))=F(g(x)) \) and
\( g'(x)=f'(x+\frac{1}{2\pi}\sin(2\pi x))(1+\frac{1}{2\pi}\cos(2\pi x)2\pi)=\underbrace{f'(x+\frac{1}{2\pi}\sin(2\pi x))}_{>0}\underbrace{(1+\cos(2\pi x))}_{\ge 0}>0 \)
For uniqueness it was Daniel's idea to consider the continuous iteration at a fixed point \( x_0 \) of \( F(x) \). The continuous iteration of \( F \) is derived from \( f \) by
(0) \( F^{\circ t}(x)=f(f^{-1}(x)+t) \)
For such an iteration to be unique it suffices to demand the existence of the limit
(1) \( \lim_{x\to x0}\frac{F^{\circ t}(x)-x_0}{x-x_0} \) for the hyperbolic fixed point or
(2) \( \lim_{x\to x0}\frac{F^{\circ t}(x)-x}{(x-x_0)^q} \) for the parabolic fixed point (where \( q>1 \) is the index of the first non-zero coefficient in the development of \( F \) at \( x_0 \)).
So (2) is our uniqueness condition for \( b=e^{1/e} \) and (1) is our uniqueness condition for \( 1<b<e^{1/e} \).
Now I was thinking further that if \( f_b(x)={}^xb \) is also analytic in \( b \), i.e. \( x\mapsto {}^t x=\exp_x^{\circ t}(1) \) is an analytic function on \( (1,e^{1/e}) \) (which has to be proved but is quite reasonable) then this function can be analytically extended to \( x\ge e^{1/e} \) if there is no singularity at \( x=e^{1/e} \).
So this would mean we had a unique analytic (in \( x \) and \( y \)) tetration \( {}^y x \) under the condition (1) and (2), where \( F \) is defined by (0).
Note, that we dont need a converging development of \( F^{\circ t}(x) \) at the fixed point \( x_0 \) (which for \( b=e^{1/e} \), \( x_0=e \), is equivalent to the existence of a converging development of \( e^x-1 \) at 0). The analytic function \( F^{\circ t}(x) \) is uniquely determined at least for \( 1<x<x_0 \) and this suffices for our \( F^{\circ t}(1) \).
Then we demand that any tetration \( f(x)={}^x b \) is a solution of the Abel equation
\( f(x+1)=F(f(x)) \) and \( f(1)=b \).
Such a solution \( f \) (even if analytic and strictly increasing) is generally not unique because for example the solution \( g(x):=f(x+\frac{1}{2\pi}\sin(2\pi x)) \) is also an analytic strictly increasing solution, by
\( g(x+1)=f(x+1+\frac{1}{2\pi}\sin(2\pi + 2\pi x))=F(f(x+\frac{1}{2\pi}\sin(2\pi x))=F(g(x)) \) and
\( g'(x)=f'(x+\frac{1}{2\pi}\sin(2\pi x))(1+\frac{1}{2\pi}\cos(2\pi x)2\pi)=\underbrace{f'(x+\frac{1}{2\pi}\sin(2\pi x))}_{>0}\underbrace{(1+\cos(2\pi x))}_{\ge 0}>0 \)
For uniqueness it was Daniel's idea to consider the continuous iteration at a fixed point \( x_0 \) of \( F(x) \). The continuous iteration of \( F \) is derived from \( f \) by
(0) \( F^{\circ t}(x)=f(f^{-1}(x)+t) \)
For such an iteration to be unique it suffices to demand the existence of the limit
(1) \( \lim_{x\to x0}\frac{F^{\circ t}(x)-x_0}{x-x_0} \) for the hyperbolic fixed point or
(2) \( \lim_{x\to x0}\frac{F^{\circ t}(x)-x}{(x-x_0)^q} \) for the parabolic fixed point (where \( q>1 \) is the index of the first non-zero coefficient in the development of \( F \) at \( x_0 \)).
So (2) is our uniqueness condition for \( b=e^{1/e} \) and (1) is our uniqueness condition for \( 1<b<e^{1/e} \).
Now I was thinking further that if \( f_b(x)={}^xb \) is also analytic in \( b \), i.e. \( x\mapsto {}^t x=\exp_x^{\circ t}(1) \) is an analytic function on \( (1,e^{1/e}) \) (which has to be proved but is quite reasonable) then this function can be analytically extended to \( x\ge e^{1/e} \) if there is no singularity at \( x=e^{1/e} \).
So this would mean we had a unique analytic (in \( x \) and \( y \)) tetration \( {}^y x \) under the condition (1) and (2), where \( F \) is defined by (0).
Note, that we dont need a converging development of \( F^{\circ t}(x) \) at the fixed point \( x_0 \) (which for \( b=e^{1/e} \), \( x_0=e \), is equivalent to the existence of a converging development of \( e^x-1 \) at 0). The analytic function \( F^{\circ t}(x) \) is uniquely determined at least for \( 1<x<x_0 \) and this suffices for our \( F^{\circ t}(1) \).