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...

Now \( slog_{\eta} ( e - \sqrt\eps) = 2e/\sqrt\eps + O(\eps) \). So we see that \( \lim_{\eps \to 0} {}^{C/\sqrt\eps}(\eta + \eps) \) is e for any constant from \( 2e \) to \( e^2/\eta \). Also, it is e for any constant less than 2e, since \( {}^{C/\sqrt\eps}(\eta) = e - 2e/{C/\sqrt\eps} + O(\eps) \). Similarly, \( \lim_{\eps \to 0} {}^{C \eps^{-1/2+\sigma}}(\eta + \eps) = e \) for any positive sigma.

But assuming positive sigma again, \( \lim_{\eps \to 0} {}^{C \eps^{-1/2-\sigma}}(\eta + \eps) \) would be \( \eta + O(\sqrt\eps) \) exponentiated \( C \eps^{-1/2-\sigma/2} - D \eps^{-1/2} \) times and then another \( C \eps^{-1/2-\sigma} - C \eps^{-1/2-\sigma/2} \); the first operation is enough to move the exponent up to at least \( e + C \eps^{-\sigma/2} \), while the next would take it from there to \( e + C/2e \eps^{-\sigma} \) and each additional step would square the excess over e again. So we would get at least \( e + C' \eps^{-\sigma 2^{C \eps^{-1/2-\sigma} - C \eps^{-1/2-\sigma/2} - 2}} \), which clearly tends to infinity as epsilon approaches zero.

So \( \lim_{\eps \to 0} {}^{C \eps^{-1/2+\sigma}}(\eta + \eps) \) is e if sigma is 0 or positive (probably independent of C), and infinite if sigma is negative.