Since you brought up the non-primary fixed points, it's worth mentioning that they aren't truly "fixed points" on the slog I've been working with. The easiest way to describe them is as fixed endpoints of loops. For example, connect a line from the second fixed point, 2.0623+7.5886*I, to its image 2.0623+1.3054*I. Now expontiate every point on the line, and you'll get a circle with endpoints at 2.0623+7.5886*I. Exponentiate again, and you'll get a complex (literally and descriptively) looping curve with endpoints at 2.0623+7.5886*I. At integer intervals, you arrive back at the same point, but continuous iteration in between is not well defined.
Going back to the regular slog as Henyrk described it, it's relatively easy to derive for bases between 1 and eta. The upper and lower fixed points give different solutions, but interestingly enough, the solution at the lower fixed point looks a heck of a lot like the logarithmic branches of bases greater than eta. The solution at the upper fixed point looks a heck of a lot like the exponential branches of bases greater than eta. I think the reason that bases greater than eta can have both types is mainly due to the complex fixed points. The regular slog at one complex fixed point or another would, for example, never go through the origin, because you can't reach it with logarithms (all fixed points are attracting when performing logarithms), and you can't reach it with exponentials (because 0 is not in the output of the exponential). But when defined, mysterious as it still is to me, as an slog using both primary fixed points, we are able to reach the origin, and indeed, this is how we know that the slog isn't the same as the regular slog as Henryk defined it, because it can be solved at the origin.
For example, the regular slog for base sqrt(2), defined for the upper fixed point at z=4, is undefined at the origin. Starting with small concentric circles centered at z=4, and exponentiating them, we eventually start to wrap around the origin. But as contorted as these wrappings may get, they never reach the origin. This is the same behavior as can be seen on the exponential branches of the slog base e, which I haven't done any pictures for yet. I suppose I should get some made up.
As far as branch cuts, I'd have to think about them more. I've been viewing it sort of as a collection of screws, with singularities as the axes of the screws, and the threads connecting to each other as one continuous sheet. I then let the branch cuts move with an "observer" embedded in the sheet, such that the branch cuts lie behind the singularities as seen by this observer. I let this "observer" wander through the slog, with branch cuts shifting as necessary so that he never sees the branch cuts, just the singularities and the two different worlds to the "left" and "right" of those singularities. I haven't given much thought to well-defined and somewhat "static" branch cuts...
Going back to the regular slog as Henyrk described it, it's relatively easy to derive for bases between 1 and eta. The upper and lower fixed points give different solutions, but interestingly enough, the solution at the lower fixed point looks a heck of a lot like the logarithmic branches of bases greater than eta. The solution at the upper fixed point looks a heck of a lot like the exponential branches of bases greater than eta. I think the reason that bases greater than eta can have both types is mainly due to the complex fixed points. The regular slog at one complex fixed point or another would, for example, never go through the origin, because you can't reach it with logarithms (all fixed points are attracting when performing logarithms), and you can't reach it with exponentials (because 0 is not in the output of the exponential). But when defined, mysterious as it still is to me, as an slog using both primary fixed points, we are able to reach the origin, and indeed, this is how we know that the slog isn't the same as the regular slog as Henryk defined it, because it can be solved at the origin.
For example, the regular slog for base sqrt(2), defined for the upper fixed point at z=4, is undefined at the origin. Starting with small concentric circles centered at z=4, and exponentiating them, we eventually start to wrap around the origin. But as contorted as these wrappings may get, they never reach the origin. This is the same behavior as can be seen on the exponential branches of the slog base e, which I haven't done any pictures for yet. I suppose I should get some made up.
As far as branch cuts, I'd have to think about them more. I've been viewing it sort of as a collection of screws, with singularities as the axes of the screws, and the threads connecting to each other as one continuous sheet. I then let the branch cuts move with an "observer" embedded in the sheet, such that the branch cuts lie behind the singularities as seen by this observer. I let this "observer" wander through the slog, with branch cuts shifting as necessary so that he never sees the branch cuts, just the singularities and the two different worlds to the "left" and "right" of those singularities. I haven't given much thought to well-defined and somewhat "static" branch cuts...
~ Jay Daniel Fox
