09/27/2007, 07:00 AM
I know I promised graphs, and I will get to them eventually, but I don't have the tools I need. I'll likely have to write a program to actually graph the data, probably using a C library to do the calculations.
In the meantime, I've been studying the slog, and I'm fairly confident of some of its basic properties. The easiest way to analyze it is to ignore for the time being any specific solution (such as Andrew's matrix solution).
Instead, imagine a half-plane, comprised of all complex values with real part less than the real part of the primary fixed points. I.e., all complex numbers with real part less than about 0.318131505. I'm going to call this half plane the "backbone" of the slog.
Now, logarithmicize this half plane. (Yes, I made that word up. I've decided it doesn't suit taking logarithms of individual values, but is an appropriate transitive verb for taking the logarithm of a set, e.g., the set of points in a defined region.)
When you do this, you will notice a few things. First, if you use only one branch when taking the logarithm of each point, you'll note that the values at the top and bottom of this 2*pi thick branch mesh with each other, including all derivatives. Hence, we can safely make copies at 2*pi*i intervals in order to analytically extend the graph. This is, in effect, filling out all the branches of the natural logarithm.
Second, there is a U-shaped region missing, centered along the real line, asymptotically approaching pi/2 above and -pi/2 below, with its smallest real part at log(0.31813) ~= -1.14529. This region corresponds to the "universe" on the other side of the line between the singularities at 0.318+1.337i and 0.318-1.337i, which we did not include. Notice that, because we made copies at 2*pi*i intervals, we'll have corresponding U-shaped cutouts.
The part of this new graph with real part greater than 0.31813, e.g., the part that has bulged between the singularities at 0.318+1.337i and 0.318+4.946i, is now in the "logarithmic" part of the slog graph. This is a logarithmic "branch" of the slog, though the slog branches in a fractal manner, so we need to be careful about what we call a logarithmic branch.
Once in this logarithmic branch, we are effectively constrained to stay in the current branch. We cannot integrate the derivative along a path to a point in a neighboring branch without hitting a "wall" singularity. We can only get to another branch by integrating a path back between the singularities through which we entered, then between the two singularities which would take us to the other point.
This is an important thing to realize, because it will prevent us, for example, from trying to use the other fixed points for continuous iteration. Doing so would require a different function than the slog we're analyzing. The slog which uses the primary fixed points for continuous iteration does not have a branch where continuous iteration from the other fixed points can be performed, unless those branches are behind the "wall" singularities, i.e., where we can't integrate a path to them, and hence we can't numerically analyze them anyway. They may as well be different functions.
What if we go the other way? That is, rather than logarithmicize the half plane, what if we exponentiate it? Well, now we get a circle with radius 1.374557..., a circle which would glance our two primary fixed points, had I not specified all complex values with real part strictly less than the real part of the fixed points.
Now we are bulging out into the "exponential" part of the slog. Note that upon entering this realm, we lose the branches. Yes, 2.062+1.305i and 2.062+7.589i both equal 2.062+7.589i when exponentiated. However, this is not due to branching at 2*pi*i intervals. Rather, we can integrate a path from 2.062+1.305i to 2.062+1.305i, and if we exponentiate the path (i.e., every point on the path), then we get a loop around the origin. The derivatives at the two points are not the same. They do not "look like" the same point. Remember, back in the "backbone", the points 0 and 2*pi*i "looked like" the same point, as far as all derivatives were concerned. In the exponential realm, two points might go to the same point when exponentiated, but the two points lie in neighborhoods with completely different properties.
(I should be careful when saying this. Yes, if we move 0.2 real units away from each point and then exponentiate, we again arrive at the same point. We might be tempted therefore to say that the neightborhoods do look the same. However, these two points are the iterative exponentiations of two different points from an underlying region. To see this, call a=2.062+1.305i and b=2.062+7.589i. Now calculate exp(ln(a)+0.1) and exp(ln(b)+0.1), and note that you arrive at different distances from a and b. Hence, the neighborhoods are not the same.)
Now, the really bizarre stuff happens when we take the second logarithm of the backbone. All of those bulges into the logarithmic branches will now appear like teeth in the formerly empty U-shaped region. In effect, the singularities at \( 0.3181315 + \left(2\pi k) \pm 1.3372357\right) i \) have now appeared within the U-shaped region, within each logarthmic branch, and all the bizarre behavior of the logarithmic and exponential branches apply as we go between these singularities as well.
All of this is done without having to know the exact mathematical "signature" of the underlying slog. However, some slogs will be "better" than others, for example, giving us smoother derivatives near the singularities. I'm hopeful that Andrew's slog will in fact be the "best" of the bunch, giving the smoothest possible derivatives near the singularities. But so far, I can at best say that his solution isn't obviously "wrong".
I also have no idea what an slog based on continuous iteration from other fixed points will look like. I may investigate it, to see if it gives me insight into this primary slog. Methinks they will have similar properties (logarithmic and exponential branches, etc.), but ultimately have different power series when constructed at the origin.
In the meantime, I've been studying the slog, and I'm fairly confident of some of its basic properties. The easiest way to analyze it is to ignore for the time being any specific solution (such as Andrew's matrix solution).
Instead, imagine a half-plane, comprised of all complex values with real part less than the real part of the primary fixed points. I.e., all complex numbers with real part less than about 0.318131505. I'm going to call this half plane the "backbone" of the slog.
Now, logarithmicize this half plane. (Yes, I made that word up. I've decided it doesn't suit taking logarithms of individual values, but is an appropriate transitive verb for taking the logarithm of a set, e.g., the set of points in a defined region.)
When you do this, you will notice a few things. First, if you use only one branch when taking the logarithm of each point, you'll note that the values at the top and bottom of this 2*pi thick branch mesh with each other, including all derivatives. Hence, we can safely make copies at 2*pi*i intervals in order to analytically extend the graph. This is, in effect, filling out all the branches of the natural logarithm.
Second, there is a U-shaped region missing, centered along the real line, asymptotically approaching pi/2 above and -pi/2 below, with its smallest real part at log(0.31813) ~= -1.14529. This region corresponds to the "universe" on the other side of the line between the singularities at 0.318+1.337i and 0.318-1.337i, which we did not include. Notice that, because we made copies at 2*pi*i intervals, we'll have corresponding U-shaped cutouts.
The part of this new graph with real part greater than 0.31813, e.g., the part that has bulged between the singularities at 0.318+1.337i and 0.318+4.946i, is now in the "logarithmic" part of the slog graph. This is a logarithmic "branch" of the slog, though the slog branches in a fractal manner, so we need to be careful about what we call a logarithmic branch.
Once in this logarithmic branch, we are effectively constrained to stay in the current branch. We cannot integrate the derivative along a path to a point in a neighboring branch without hitting a "wall" singularity. We can only get to another branch by integrating a path back between the singularities through which we entered, then between the two singularities which would take us to the other point.
This is an important thing to realize, because it will prevent us, for example, from trying to use the other fixed points for continuous iteration. Doing so would require a different function than the slog we're analyzing. The slog which uses the primary fixed points for continuous iteration does not have a branch where continuous iteration from the other fixed points can be performed, unless those branches are behind the "wall" singularities, i.e., where we can't integrate a path to them, and hence we can't numerically analyze them anyway. They may as well be different functions.
What if we go the other way? That is, rather than logarithmicize the half plane, what if we exponentiate it? Well, now we get a circle with radius 1.374557..., a circle which would glance our two primary fixed points, had I not specified all complex values with real part strictly less than the real part of the fixed points.
Now we are bulging out into the "exponential" part of the slog. Note that upon entering this realm, we lose the branches. Yes, 2.062+1.305i and 2.062+7.589i both equal 2.062+7.589i when exponentiated. However, this is not due to branching at 2*pi*i intervals. Rather, we can integrate a path from 2.062+1.305i to 2.062+1.305i, and if we exponentiate the path (i.e., every point on the path), then we get a loop around the origin. The derivatives at the two points are not the same. They do not "look like" the same point. Remember, back in the "backbone", the points 0 and 2*pi*i "looked like" the same point, as far as all derivatives were concerned. In the exponential realm, two points might go to the same point when exponentiated, but the two points lie in neighborhoods with completely different properties.
(I should be careful when saying this. Yes, if we move 0.2 real units away from each point and then exponentiate, we again arrive at the same point. We might be tempted therefore to say that the neightborhoods do look the same. However, these two points are the iterative exponentiations of two different points from an underlying region. To see this, call a=2.062+1.305i and b=2.062+7.589i. Now calculate exp(ln(a)+0.1) and exp(ln(b)+0.1), and note that you arrive at different distances from a and b. Hence, the neighborhoods are not the same.)
Now, the really bizarre stuff happens when we take the second logarithm of the backbone. All of those bulges into the logarithmic branches will now appear like teeth in the formerly empty U-shaped region. In effect, the singularities at \( 0.3181315 + \left(2\pi k) \pm 1.3372357\right) i \) have now appeared within the U-shaped region, within each logarthmic branch, and all the bizarre behavior of the logarithmic and exponential branches apply as we go between these singularities as well.
All of this is done without having to know the exact mathematical "signature" of the underlying slog. However, some slogs will be "better" than others, for example, giving us smoother derivatives near the singularities. I'm hopeful that Andrew's slog will in fact be the "best" of the bunch, giving the smoothest possible derivatives near the singularities. But so far, I can at best say that his solution isn't obviously "wrong".
I also have no idea what an slog based on continuous iteration from other fixed points will look like. I may investigate it, to see if it gives me insight into this primary slog. Methinks they will have similar properties (logarithmic and exponential branches, etc.), but ultimately have different power series when constructed at the origin.
~ Jay Daniel Fox
