Periodic analytic iterations by Riemann mapping

[tommy1729]

Let r denote riemann mapping.
r' denotes the functional inverse of r.

Let j be An analytic jordan curve around the origin.

R maps j to the unit circle.

Let r'(1) be the starting point A.

then iterations f^[x](r'(1)) with x > 0 ( real ) are periodic and analytic iff

F(z) = r' ( T ( r) ).
Where T is a complex number on the unit circle.

I think this is correct.

This seems to suggest exp has no periodIC iterational Jordan curve away from its fixpoints.

Regards

Tommy1729