Infinite tetration of the imaginary unit

[tommy1729]

sheldon , we dont know if the period is a brjuno number , which is why we are not certain if its a siegel disk.

( else i would have posted it myself )

i dont know much about brjuno numbers but it seems deciding if a algebraic number of degree 3 is a brjuno number is already partially unresolved , hence i doubt period 2.98... is easy.

( i also dont know much about herman rings , feel free to inform me )

although you are correct with probability 1 , we know how " tricky and weird " tetration can be.

i would also like to point out that 1.7129i^z is periodic with 2pi i /ln(1.7129 i) = 3.57977339 + 1.22650561 i ( another mysterious number ) , such that 1.7129i^(z + 3.57977339 + 1.22650561 i) ends up in the same orbit as 1.7129i^z and this probably defines the boundary of the siegel disk ( because of the univalent riemann mapping ) and settles my question about large initial values like 1729 + 1729 i.

so we have at least 2 types of periods : 2.98... and 3.57977339 + 1.22650561 i that dominate the bahaviour of the superfunction sexp_1.7129i(z) on the main branch.

sexp_1.7129i(z) will have its fixpoints (of 1.7129i^z) at +/- oo i.

i dare to consider ( not yet conjecture :p )

e§ = 1.7129i^z ln§ = ln(z) / ln(1.7129i) 2§ = e§ - 1/e§

for real x and some suitable real constant c

sexp_1.7129i(x) = ln§ ln§ ln§ ... 2§^[x]( e§ e§ e§ ... © )

- with the correct branches - in analogue to my 2sinh method.

( plz dont copy the same arguments as in the sinh thread if irrelevant )

regards

tommy1729