From Osher Doctorow (Marlene is the network listname owned by my wife Marleen (notice the spellings) and I will eventually get my own user name on this network).

Readers might be interested in some of the postings that I've made to http://www.mathforum.org/epigone/geometry-research and other places concerning the relationships among chaos, Rare Event Theory (RET), and pathology for functions with real ranges and real domains. To summarize in more or less ordinary English, chaos appears to derive from "pathologies" in the REAL numbers even though complex numbers are typically used for fractals and chaos. The intuitive idea of a pathology as something "ill" or "sick" or "life-threatening" is pretty close to it. Of course, that is relative to the system that one is considering - to a virus, maybe we look like pathologies. But we usually consider that life is the "non-ill" system, for better or worse, and THINGS WHICH CONTRIBUTE TO THE SYSTEM are non-pathological, while THINGS WHICH DETRACT FROM THE SYSTEM are pathological. People familiar with engineering systems may quickly get this idea. It turns out in Rare Event Theory (RET) that Rare Events (those with probability less than .05 or .01 typically, although .10 is occasionally used and there are arguments for using as much as 1/3 or even .50 in some circumstances) follow the pattern of the function p1 (x, y) = 1 + y - x with y < = x for their "non-pathological" probable influence behavior, with x and y usually taken between 0 and 1. Notice carefully that p1 is 1 if and only if (iff for short) y = x, which includes points like (0, 0), (1, 1), (x, x) for x between 0 and 1. The first number or letter in (0, 0) refers to the x value, the second to the y value, and similarly for the others. Now let's look at the typical chaotic Logistic map or equation (chaotic for certain values and intervals): 1) y = Lx(1 - x) where L is a constant (fixed real number) greater than 0 here. A closely related equation is the logistic function equation: 2) y' = kx(a - x) where a, k are constants with a > 0 and y' is the growth of y or technically rate of change of y at a particular point x which in the case of (2) is usually taken as x = t (time). Equation (2) is often used to model biological/population growth. Let's change a to 1 for simplicity (we can always make a scale change for example, although when the magnitude of a is studied we'd want to change it back). The equation y = Lx(1 - x) contains two variable factors, x and 1 - x. We can fairly easily see that x is non-pathological and 1 - x is pathological by considering that if x is mass divided by the total mass of the galaxy assumed constant or population divided by maximum supportable population (which is a fraction between 0 and 1) for example, then life increases with x but decreases with 1 - x. It also turns out that (0, 0) and (1, 1) are on y = x, but not on y = 1 - x. In fact, if x = 0, then 1 - x = 1, and if x = 1, then 1 - x = 0. So only (1, 0) is on y = 1 - x (remember that y < = x so y can't be bigger than x). That's all there is to it in essence except to note that for certain values of L and for certain ranges of x relative to this value, things get chaotic. Either equation (1) or (2) is a combination of two conflicting influences - non-pathological and pathological. That's where the chaos comes from when it does occur. I didn't say anything about complex numbers of form x + iy here, where i is the square root of -1. It turns out that many ideas in complex numbers including the complex conjugate x - iy of x + iy extend to the real numbers provided that we stick to Rare Events. How much confidence can readers have in these results in view of the fact that I'm just developing them now? Well, take a look at my postings at the above site, which extend back several years, and which refer also to some of my publications, and also my postings at http://www.superstringtheory.com/forum in the String/M Theory/Duality subforum of their Forum section, and if necessary I'll cite some of my publications directly. But pathology is already well known in super- abstract algebra - it's called the Jacobson Radical (look it up as an internet keyword) which has the star product (look that up) x o y = x + y - xy. Osher Doctorow Ph.D. One or More of California State Universities and Community Colleges