possible tetration extension part 3

[bo198214]

seems like using the typical dexp(x) = exp(x) - 1 in the usual way.

No, actually it is rather similar to your 2sinh method!

The general principle is the following: Say we have some flow \(f^{s+t}(x)=f^s(f^t(x))\)
And the limit 
\[ F^t(x)=\lim_{n\to\infty} \log_b^{\circ n}(f^t(\exp_b^{\circ n}(x))) \] 
exists then \(F^t\) is again a flow, because roughly (this is not a proof):
\begin{align}
F^s(F^t(x)) &= \lim_{n\to\infty} \log_b^{\circ n}(f^s(\exp_b^{\circ n}(\log_b^{\circ n}(f^t(\exp_b^{\circ n}(x))))))\\
                  &= \lim_{n\to\infty} \log_b^{\circ n}(f^{s+t}(\exp_b^{\circ n}(x)))\\
                  &= F^{s+t}(x)
\end{align}

You use the regular iterates of \(f(x)=2\sinh(x)\) as flow, while Shanghai46 uses the regular iterates of \(f(x)=b^x-1\) as flow. Because both functions satisfy (if one extends sinh to other bases accordingly):
\begin{align}
\lim_{n\to\infty} \log_b^{\circ n}(f(\exp_b^{\circ n}(x))) &= \exp_b(x) \\
F^{t+1}(x) &= \exp_b(F^t(x))
\end{align}

And this is the main dilemma with those approaches they are just countless!

I think also Peter Walker used a similar approach.