Phew, lots of questions. But regarding the dependency from a specific slog, there surely is one regarding the slog derivative (which is quite clear) and regarding the equilibrium:
For example if we consider the just discussed regular slog at the lower real fixed point \( a \)
\( \alpha_b(x)=\log_{\ln(a)}\left(\lim_{n\to\infty} \frac{\exp_b^{\circ n}(x)}{\ln(a)^n}\right) \), \( \text{rslog}_b(x)=\alpha_b(x)-\alpha_b(1)=\log_{\ln(a)}\left(\lim_{n\to\infty}\frac{\exp_b^{\circ n}(x)}{\exp_b^{\circ n}(1)}\right) \)
there is a dependency of the real part from the imaginary argument part.
For example for \( b=\sqrt{2},a=2 \):
\( \text{rslog}_b(1)=0 \) and \( \text{rslog}_b(1+\frac{I}{2})=-0.2625038052+0.7542335672*I \)
For example if we consider the just discussed regular slog at the lower real fixed point \( a \)
\( \alpha_b(x)=\log_{\ln(a)}\left(\lim_{n\to\infty} \frac{\exp_b^{\circ n}(x)}{\ln(a)^n}\right) \), \( \text{rslog}_b(x)=\alpha_b(x)-\alpha_b(1)=\log_{\ln(a)}\left(\lim_{n\to\infty}\frac{\exp_b^{\circ n}(x)}{\exp_b^{\circ n}(1)}\right) \)
there is a dependency of the real part from the imaginary argument part.
For example for \( b=\sqrt{2},a=2 \):
\( \text{rslog}_b(1)=0 \) and \( \text{rslog}_b(1+\frac{I}{2})=-0.2625038052+0.7542335672*I \)