Theorem on increment of tetration

Theorem on increment of superfunctionsis statement about asymptotic behavior of solution of the Transfer equation.

Let \(F\) be solution of equation

\(F(z\!+\!1)=\exp\big(\beta F(z)\big)\)

for some \(\beta>0\).

Let \(L\) be the fixed point, id est, \(\exp(\beta L)=L\)

Let \(F(z)=L+\varepsilon+O(\varepsilon^2) \)

where \(\varepsilon = \exp(kz) \) for some increment \(k\).

Let \(~ K\!=\!\exp(k)\)

Then

\( \Im(K) = \Im(k) \)