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 Does tetration take the right half plane to itself? JmsNxn Long Time Fellow Posts: 481 Threads: 86 Joined: Dec 2010 05/16/2017, 04:46 AM (This post was last modified: 05/16/2017, 04:59 AM by JmsNxn.) 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. « Next Oldest | Next Newest »

 Messages In This Thread Does tetration take the right half plane to itself? - by JmsNxn - 05/10/2017, 07:46 PM RE: Does tetration take the right half plane to itself? - by JmsNxn - 05/10/2017, 08:22 PM RE: Does tetration take the right half plane to itself? - by sheldonison - 05/15/2017, 08:16 PM RE: Does tetration take the right half plane to itself? - by sheldonison - 05/15/2017, 09:00 PM RE: Does tetration take the right half plane to itself? - by JmsNxn - 05/16/2017, 04:09 AM RE: Does tetration take the right half plane to itself? - by sheldonison - 05/16/2017, 03:34 PM RE: Does tetration take the right half plane to itself? - by JmsNxn - 05/16/2017, 08:46 PM RE: Does tetration take the right half plane to itself? - by JmsNxn - 05/16/2017, 04:46 AM

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