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The fractal nature of iterated ln(x) [Bandwidth warning: lots of images!]
#11
Argh, that's embarrassing... I was right in some respects, but I made a key error in my thought experiment. Large values will tend towards 0 with high probability (essentially 50% once the imaginary part gets large), and iterating values close to zero will tend to resemble iteration with real numbers (i.e., integer tetrations of e). Eventually, an infinitesimal imaginary part will get large enough to cause a repeat. However, with very large numbers, the next time through the cycle, we'll be even closer to 0, so that the iterates further resemble real numbers, and we have to exponentiate even further before the imaginary part becomes large enough to send us back even closer to 0, and so on. If taken to its logical conclusion, we should be able to eventually recover any number of consecutive integer tetrations of e with arbitrary precision, by iteratively exponentiating a random complex number.

In short, if you exponentiate a complex number enough times, it should eventually reach a point where it's essentially a real number. Of course, exponentiate it enough times further, and it will get "sent" back to an infinitesimal value near 0, where the process repeats (and goes a little further the time through).

And on the other hand, if you take iterated logarithms of any complex number, you should eventually converge on a fixed point. Assuming you use the principal branch, you should settle on the primary fixed point (or its conjugate). As far as I can tell, the other fixed points require use of different branches of the logarithm.

And, having been wrong already, I reserve the right to be wrong again...
~ Jay Daniel Fox
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#12
jaydfox Wrote:In short, if you exponentiate a complex number enough times, it should eventually reach a point where it's essentially a real number. Of course, exponentiate it enough times further, and it will get "sent" back to an infinitesimal value near 0, where the process repeats (and goes a little further the time through).

And on the other hand, if you take iterated logarithms of any complex number, you should eventually converge on a fixed point. Assuming you use the principal branch, you should settle on the primary fixed point (or its conjugate). As far as I can tell, the other fixed points require use of different branches of the logarithm.
For me, when I played around with this, I got the impression, that it might be best expressed, that the (attracting) fixed point has a gradient-field around it. In polar notation, (where the fixed point is translated to the origin), the lengthes and the angle from one iteration to the next converged to a certain value for each starting point (and also for each fixed point). So the inverse process, starting at the fixed point to arrive at the starting point means to select a direction and a length-change - inexpressible for me in terms of the infinitesimal changes... But I remember the term "gradient field" from my "Hütte - mathematische Tabellen" and it seems to fit here.

Gottfried
Gottfried Helms, Kassel
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#13
What I find strange is that it seems that the iterated logarithm of every point in the upper complex half plane tends to the one fixed point in the upper plane, though there are many other attracting fixed points of in the upper half plane. Similarly for the lower half plane.
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#14
bo198214 Wrote:What I find strange is that it seems that the iterated logarithm of every point in the upper complex half plane tends to the one fixed point in the upper plane, though there are many other attracting fixed points of in the upper half plane. Similarly for the lower half plane.

I could be wrong, but I think there's only one true fixed point per branch. The others are "images" of it from other branches of the natural logarithm.

For example, if you always use the branch with imaginary part in the range , then your fixed point is 2.06227773+7.588631178i.

Now, if you take the natural logarithm of that point with the principle branch, you get 2.06227773+1.305445871i. So that point would appear to be a fixed point of exponentiation. If you exponentiate it, you get back to 2.06227773+7.588631178i, and the next exponentiation stays there, using that point as a fixed point.

Therefore, there are many fixed points of exponentiation, but only one fixed point per branch of the natural logarithm (two if you count the conjugate as a separate value).
~ Jay Daniel Fox
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#15
I wonder, however, if there are "mixed" fixed points. You know, alternate between two branches for every other logarithm, or three branches for every third, etc. What type of values do we settle on?
~ Jay Daniel Fox
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#16
jaydfox Wrote:I wonder, however, if there are "mixed" fixed points. You know, alternate between two branches for every other logarithm, or three branches for every third, etc. What type of values do we settle on?

I dont think so. We have in each rectangle of values with , , exactly one fixed point and these together with its conjugates are all fixed points of .
I think the fixed point with imaginary part in is the limit of the iterated for a starting value with positive imaginary part.
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#17
bo198214 Wrote:
jaydfox Wrote:I wonder, however, if there are "mixed" fixed points. You know, alternate between two branches for every other logarithm, or three branches for every third, etc. What type of values do we settle on?

I dont think so. We have in each rectangle of values with , , exactly one fixed point and these together with its conjugates are all fixed points of .
I think the fixed point with imaginary part in is the limit of the iterated for a starting value with positive imaginary part.

Sorry, I think I wasn't clear. I was thinking that if we alternated logarithms between two branches, we should settle on two points (one for every other iteration) that are "fixed", such that if you exponentiated, you would go back and forth between two points, until you'd moved sufficiently far to escape. Similarly, three points for three-cycle iterated logarithms, etc.

At any rate, while fascinating (and something I may study further in the future), I think I've drifted far enough away from my original line of inquiry that its time to start steering back on course. All of this study is to get a better idea of how iterated exponentiation "works", so that I can figure out how to derive a continuous solution based on principles of exponentiation, rather than from Andrew's sneaky trick of solving a linear system of equations based on the inverse of iterated exponentiation. Unforunately, the best I've found so far is a relation between the primary fixed point of natural logarithms and the coefficients of the power series of Andrew's slog for base e.

It is my hope to find a direct solution for base e, have it be reasonably defensible, and hopefully have it match either Andrew's solution or mine (preferably Andrew's).
~ Jay Daniel Fox
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