base holomorphic tetration

[bo198214]

Give me the weekend and I will make a complex contour plot of some regular tetrapower

Here are finally the pictures (damn did that take time, as a side effect I discovered a bug in the python mpmath.lambertw function). I show the conformal map of z[4]0.5 inside the upper half of the Shell-Thron region via regular iteration, i.e. \( z[4]0.5 = \exp_b^{\circ 0.5}(1) \).

       

For the accuracy I checked the difference \( \exp_b^{\circ 1/2}(\exp_b^{\circ 1/2}(1))- b \). It is always under \( 10^{-8} \).

I computed the regular iteration as a mixture of the powerseries development at the lower fixed point, together with (if 1 is not inside the convergence radius of this powerseries) the equality \( \log_b^{\circ n} \circ \exp_b^{\circ t} \circ \exp_b^{\circ n} = \exp_b^{\circ t} \), i.e. the exponentials move the argument towards the fixed point until it is inside its convergence radius then the powerseries can be applied and the same number of logarithms have to finish the computation.

I assumed the convergence radius to be at least the distance of the fixed point from the image of the Shell-Thron region.