09/07/2007, 09:13 PM
In fact, now that I think about it, if you pick any complex number at random, the probability is 1 (i.e., almost certain) that if you iteratively exponentiate it, you will eventually cover the entire complex plane in a dense set of measure 0.
There is a small chance (probability of 0) that you will end up on a real number after some finite number of iterations, after which, of course, this behavior ceases. In most cases, however, a small imaginary part will grow exponentially until it reaches a value very near (k+1/2)*pi, in which case the next iteration will become a large imaginary number with small real part, and the next iteration will tend towards a root of unity. Just as likely is that the imaginary part will grow to (2k+1)*pi, in which case the next iteration will tend towards a large negative number, and the next iteration will tend to 0. If the imaginary part approaches 2k*pi, then the next iteration will have a small imaginary part, which brings us back to the beginning of this thought exercise.
These cycles are what allow us to eventually cover the entire complex plane in most cases. There are loops which repeat with a certain periodicity (every 2nd iteration, or every 3rd, or every 4th, etc.), but these will be unstable because the fixed points are repelling.
This assumes that there aren't any attracting fixed points. I haven't found any yet, but I haven't been looking for them specifically. Has anyone else found attracting fixed points?
There is a small chance (probability of 0) that you will end up on a real number after some finite number of iterations, after which, of course, this behavior ceases. In most cases, however, a small imaginary part will grow exponentially until it reaches a value very near (k+1/2)*pi, in which case the next iteration will become a large imaginary number with small real part, and the next iteration will tend towards a root of unity. Just as likely is that the imaginary part will grow to (2k+1)*pi, in which case the next iteration will tend towards a large negative number, and the next iteration will tend to 0. If the imaginary part approaches 2k*pi, then the next iteration will have a small imaginary part, which brings us back to the beginning of this thought exercise.
These cycles are what allow us to eventually cover the entire complex plane in most cases. There are loops which repeat with a certain periodicity (every 2nd iteration, or every 3rd, or every 4th, etc.), but these will be unstable because the fixed points are repelling.
This assumes that there aren't any attracting fixed points. I haven't found any yet, but I haven't been looking for them specifically. Has anyone else found attracting fixed points?
~ Jay Daniel Fox