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 \( 0 < \lambda < 1 \), if \( \lambda^{z}: \mathbb{C}_{\Re(z)>0} \to \mathbb{D} \) (where \( \mathbb{D} \) 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 \( \Re(z) \to \infty \), and is INJECTIVE modulo its period.
We get the exact same thing with tetration \( ^z\alpha \), substituting \( 0 = L \) and \( \lambda = \log(L) \) and \( \mathbb{D} = \{z \in \mathbb{C} : |z-L| \le |L-1| \). 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 \( \lambda^z \) on some weird simply connected domain.
Let \( 0 < \lambda < 1 \), if \( \lambda^{z}: \mathbb{C}_{\Re(z)>0} \to \mathbb{D} \) (where \( \mathbb{D} \) 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 \( \Re(z) \to \infty \), and is INJECTIVE modulo its period.
We get the exact same thing with tetration \( ^z\alpha \), substituting \( 0 = L \) and \( \lambda = \log(L) \) and \( \mathbb{D} = \{z \in \mathbb{C} : |z-L| \le |L-1| \). 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 \( \lambda^z \) on some weird simply connected domain.