Rereading your second post, this much can be proved VERY easily! The fact that tetration cycles around the fixed point can exactly be related to the exponential map.

Let , if (where is the unit disk), It is periodic (first of all) and rotates around zero as we increase the imaginary argument, it tends to zero as , and is INJECTIVE modulo its period.

We get the exact same thing with tetration , substituting and and . This is biholomorphic to the previous scenario via the Schroder function. It's no surprise that it behaves how you described. I always wondered what kind of crazy fractals it performed when it cycled around, but that much I couldn't plot. I just knew that many fractional iterations created a weird on some weird simply connected domain.

Let , if (where is the unit disk), It is periodic (first of all) and rotates around zero as we increase the imaginary argument, it tends to zero as , and is INJECTIVE modulo its period.

We get the exact same thing with tetration , substituting and and . This is biholomorphic to the previous scenario via the Schroder function. It's no surprise that it behaves how you described. I always wondered what kind of crazy fractals it performed when it cycled around, but that much I couldn't plot. I just knew that many fractional iterations created a weird on some weird simply connected domain.