Deriving tetration from selfroot?

[Ivars]

I tried to iterate infinite tetration h in both negative and positive directions and it seems that :

if h( z^1/z)) = z = then first negative iteration of h is z^(1/z), second

(z^(1/z))^(1/(z^(1/z)) and so on.

negative infinite iteration of (i^(1/i), but also then (i) also then 0,438283+0.3605924*i etc leads to 1
negative infinite iteration of h(e^(1/e)) then also e leads to 1,

Positive infinite iteration of h (i) -> h(h(h(h(......h(i))= 1 ( I only checked 30 iterations).

Does it mean that all infinite positive negative iterations of h(z) lead to 1?

Ivars