I'm still working on some graphs, but to give a sneak preview, here's a graph of \( \exp^{\circ z}(0) \), focussing mainly on complex iterations with real part between -4.5 and 1.5, imaginary part between 0i and 1.25i. The blue region is the rectangle with upper left corner -0.5+1.25i, lower right corner 0.5+0i. Each color is an integer real iterate of this "critical" interval. (A true critical interval would only go up about 1.0579i, but I don't have enough terms in Andrew's slog to try to get that fancy yet, and besides, I wanted to show some overlap.)
The dark lines represent integer real and integer imaginary iterations. The medium lines are quarter-real and quarter-imaginary iterates. The faint lines are 1/20th iterations (real and imaginary). The main reason for iterating 0 rather than 1 is because the radius of convergence of Andrew's slog limits me to this range. (The radius of convergence is limited by the singularity at the fixed point.)
As you can hopefully get a feel for, using iterated logarithms alone is a hopeless approach, as I've already demonstrated in the fractal nature of iterated ln(x) discussion. You cannot continuously iterate between two successive iterates, despite the fact that the integer iterates of any particular real number will behave as if they were part of an exponential complex spiral.
However, from this graph, we can see why the key to Andrew's solution is complex iterations. Start from any real number, preferably a number between 0 and 1. The slope of Andrew's slog determines how "fast" we move towards larger real numbers. If you go out at a right angle at that same "speed", you start heading towards the fixed point.
By the time you get to the ith iteration, you have a very nice, smooth curve. Notice the spiky nature of the region around \( \exp^{\circ -4}(0) \), versus the very smooth nature of the region around \( \exp^{\circ -4+i}(0) \). On the graph, The pointy tip of the red region is \( \exp^{\circ -4}(0) \), while the dark red "cross" is where \( \exp^{\circ -4+i}(0) \) is located.
I therefore hypothesize that if you take a large number of imaginary iterates, you'll get very close to the fixed point, and more importantly, using exponentiation will become increasingly valid, i.e., after using negative imaginary iterates to climb back out from the fixed point, you'll be closer to the real line.
The only problem is, how do we define imaginary iterates? Well, with Andrew's slog-based solution, the imaginary iterates seem to be well-behaved. The "correct" solution will in fact be very well-behaved. A solution that is off by some cyclic amount will have complex iterates that are not quite "parallel" (not the right word, but...), so that they will tend to bunch up as we approach the fixed point. I plan to test this with my solution, and show the difference, so that you can get a feel for what I mean by this.
The dark lines represent integer real and integer imaginary iterations. The medium lines are quarter-real and quarter-imaginary iterates. The faint lines are 1/20th iterations (real and imaginary). The main reason for iterating 0 rather than 1 is because the radius of convergence of Andrew's slog limits me to this range. (The radius of convergence is limited by the singularity at the fixed point.)
As you can hopefully get a feel for, using iterated logarithms alone is a hopeless approach, as I've already demonstrated in the fractal nature of iterated ln(x) discussion. You cannot continuously iterate between two successive iterates, despite the fact that the integer iterates of any particular real number will behave as if they were part of an exponential complex spiral.
However, from this graph, we can see why the key to Andrew's solution is complex iterations. Start from any real number, preferably a number between 0 and 1. The slope of Andrew's slog determines how "fast" we move towards larger real numbers. If you go out at a right angle at that same "speed", you start heading towards the fixed point.
By the time you get to the ith iteration, you have a very nice, smooth curve. Notice the spiky nature of the region around \( \exp^{\circ -4}(0) \), versus the very smooth nature of the region around \( \exp^{\circ -4+i}(0) \). On the graph, The pointy tip of the red region is \( \exp^{\circ -4}(0) \), while the dark red "cross" is where \( \exp^{\circ -4+i}(0) \) is located.
I therefore hypothesize that if you take a large number of imaginary iterates, you'll get very close to the fixed point, and more importantly, using exponentiation will become increasingly valid, i.e., after using negative imaginary iterates to climb back out from the fixed point, you'll be closer to the real line.
The only problem is, how do we define imaginary iterates? Well, with Andrew's slog-based solution, the imaginary iterates seem to be well-behaved. The "correct" solution will in fact be very well-behaved. A solution that is off by some cyclic amount will have complex iterates that are not quite "parallel" (not the right word, but...), so that they will tend to bunch up as we approach the fixed point. I plan to test this with my solution, and show the difference, so that you can get a feel for what I mean by this.
~ Jay Daniel Fox