tetration limit ??

[BenStandeven]

Let's see here.

The fixed point for base (eta + eps) is e + delta(eps), where delta satisfies:

\( \delta(\eps) = -e^2 \eps/\eta + \sqrt{2/\eta} e^{3/2} \sqrt{-\eps} + O(\eps^{3/2}) \)

\( (\eta + \eps)^{e + \Re(\delta(\eps)) + \theta} = e^{1 + \theta (1 + e \eps/\eta)/e} + O(\eps^{3/2}) \)

Now if \( \theta \) is on the order of \( \sqrt \eps \), we have \( (\eta + \eps)^{e + \Re(\delta(\eps)) + \theta} = e + \theta + \theta^2/2e + O(\eps^{3/2}) \), so the effect of an additional level of tetration is to add \( -\Re(\delta(\eps)) + \theta^2/2e \) to the exponent. To cross this region of length \( 2 \sqrt \eps \) would require between \( 2 / (\sqrt \eps (e^2/\eta + 1/2e)) \) and \( 2 / (\sqrt \eps (e^2/\eta)) \) steps.

But if \( \theta \) is of a larger order, the epsilon-dependent terms may be neglected, and we get that \( (\eta + \eps)^{e + \theta} = \eta^{e + \theta} + O(\eps)} \).

So it takes roughly \( slog_{\eta}(e - \sqrt\eps) \) tetration levels to reach \( e - \sqrt \eps \).

To be continued...