Continuous iteration from fixed points of base e

[jaydfox]

By fixed points, I'm referring to picking one and only one branch of the logarithm, and iterating until you settle on a single value. There is one such value per branch, though they come in conjugate pairs, so the principal branch has two.

Anyway, the a_k are the "upper" fixed points of iteration for \( \ln(z)+2\pi k i \), with k >= 0. Then the conjugates define the "lower" fixed points.

The first few, enumerated to clear up any potential doubt, are:

Code:

k | a_k
0 | 0.318131505... + 1.337235701...i
1 | 2.06227773... + 7.588631178...i
2 | 2.653191974... + 13.94920833...i
3 | 3.020239708... + 20.27245764...i

Now if you alternate branches, there are fixed cycles. The simplest are the conjugating cycles, such as 1.668024052+5.032447064...i. When exponentiated, it simply gets conjugated. I didn't include these or any of the other cycles. There are cycles of all integer lengths. I don't know how any of these affect the slog, and I assume they don't. They would be in other "branches" of the slog not easily accessible from the origin.