bo198214 Wrote:\( {\exp_b}^{\circ t}(x)=\lim_{a\downarrow \eta} \lim_{n\to\infty} {\log_b}^{\circ n}(\text{sexp}_a (t+\text{slog}_a({\exp_b}^{\circ n}(x)))) \)
I wonder whether this formula works also if we dont take the limit \( a\downarrow \eta \) but directly put:
\( {\exp_b}^{\circ t}(x)=\lim_{n\to\infty} {\log_b}^{\circ n}(\text{sexp}_\eta (t+\text{slog}_\eta({\exp_b}^{\circ n}(x)))) \)
where of course we can again replace
\( \text{sexp}_\eta(t+\text{slog}_\eta(y)))={\exp_\eta}^{\circ t}(y) \)
where we can compute \( {\exp_\eta}^{\circ t} \) by regular iteration because it has a fixed point at \( e \) (though not with Zdun's formula as his formula is only valid for \( |q|<1 \) while \( {\exp_\eta}'(e)=1 \)).
