using sinh(x) ?

[bo198214]

I followed most of that, what is the constant value? the upper fixed point of a?

Yes I write \( a^+ \) for the upper fixed point of \( a^x \) and \( a^- \) for the lower fixed point of \( a^x \) if \( 1<a\le e^{1/e} \).

Also, what is the circle \( \circ \) notation, \( \mu_{a,b}\circ\exp_b = \exp_a\circ\mu_{a,b} \)? Is it the same as \( \mu_{a,b}(\exp_b) = \exp_a(\mu_{a,b}) \)? No...

Yes \( f\circ g \) is the function given by \( (f\circ g)(x)=f(g(x)) \).
\( \circ \) is called the composition operator.
Thatswhy you see sometimes in literature \( f^{\circ n} \), which means the \( n \)-th power of \( f \), but not the power for the multiplication operation, but the power for the composition operation, i.e. \( f^{\circ n}=f\circ \dots\circ f \).
The notation \( f(g) \) is formally a bit arguable because \( g \) is not from the domain of definition of f, like it is if you write \( f(x) \).

[quote][quote]
But even then:
If \( \sigma_a \) and \( \sigma_b \) are the regular superfunctions to base \( a \) and \( b \) respectively at the *upper* fixed point, then
\( \mu_{a,b}\circ \sigma_b \) is a superfunction to base \( a \) but
\( \mu_{a,b}\circ \sigma_b \neq \sigma_a \).

Yeah, I would expect all of the same problems of iterating complex logarithms, with super exponential windings, and singularities.

I just wanted to point out that
regular iteration at \( \eta=e^{1/e} \) basechange to \( e \) does not give Kouznetsov's (but wobbles)
regular iteration at \( a<\eta \) basechange to \( b<\eta \) does not give regular iteration (but wobbles). (though regular iteration must be considered always at the upper fixed point, not as we are used to at the lower fixed point.)