I can't be the first to have made this observation, so I was wondering if anyone knew of any websites devoted to exploring this fascinating topic. I know it's a bit off topic, but not by too much.
In thinking about it, as we iterate a complex number, its modulus grows exponentially so long as the argument is close to 0. Eventually, the imaginary part will grow sufficiently to boost it up to unit sized or bigger.
Assume a random input. After several exponentiations, assume we have a number with a "large" real part and "small" imaginary part. For example, 123.45 + 0.00012345i. The next iterate will therefore have a "large" imaginary part and a "very large" real part. The "large" imaginary part is essentially i times a random number centered at zero but possibly several hundred (or several trillion, etc.) multiples of pi times i. Therefore, there is essentially a 50% chance that it's imaginary part is between \( (2k-{\small \frac{1}{2}})\pi \) and \( (2k+{\small \frac{1}{2}})\pi \), in which case the next iterate will have a larger modulus, and again the imaginary part is large and essentially random.
There is essentially a 50% chance that the imaginary part is in the other range, which produces complex numbers with negative real part. Regardless of the magnitude of the real and imaginary parts, the next iteration will necessarily be within a unit circle, with a high probability of being close to 0.
This behavior is fascinating! Once an iterate gets "large" (greater than 5, greater than 100, take your pick), each successive iterate has a 50% chance of continuing to get larger, or a 50% of shrinking back towards 0, where we repeat this process. Since these numbers will act randomly (assuming a random input), we should cover all complex numbers eventually, though the numbers less than 1 are far more likely to occur, and the numbers close to fixed points are also far more likely to occur.
The question is, can we graphically show how likely each point in the complex plane is to appear in such an iteration sequence? Assuming a random input gives essentially random outputs, it shouldn't matter what the input value is (except in the 0% chance that we pick a point that loops or eventually becomes real). I'm thinking that a logarithmic (or superlogarithmic) scaling of the probability would help bring out the details.
We wouldn't need a whole lot of precision in our math library, because once we get into the range of "large" numbers, if the exponentiation exceeds the precision of our library, we could just pick a random modulus with an appropriate distribution around 0, and a random argument, and start over.
In thinking about it, as we iterate a complex number, its modulus grows exponentially so long as the argument is close to 0. Eventually, the imaginary part will grow sufficiently to boost it up to unit sized or bigger.
Assume a random input. After several exponentiations, assume we have a number with a "large" real part and "small" imaginary part. For example, 123.45 + 0.00012345i. The next iterate will therefore have a "large" imaginary part and a "very large" real part. The "large" imaginary part is essentially i times a random number centered at zero but possibly several hundred (or several trillion, etc.) multiples of pi times i. Therefore, there is essentially a 50% chance that it's imaginary part is between \( (2k-{\small \frac{1}{2}})\pi \) and \( (2k+{\small \frac{1}{2}})\pi \), in which case the next iterate will have a larger modulus, and again the imaginary part is large and essentially random.
There is essentially a 50% chance that the imaginary part is in the other range, which produces complex numbers with negative real part. Regardless of the magnitude of the real and imaginary parts, the next iteration will necessarily be within a unit circle, with a high probability of being close to 0.
This behavior is fascinating! Once an iterate gets "large" (greater than 5, greater than 100, take your pick), each successive iterate has a 50% chance of continuing to get larger, or a 50% of shrinking back towards 0, where we repeat this process. Since these numbers will act randomly (assuming a random input), we should cover all complex numbers eventually, though the numbers less than 1 are far more likely to occur, and the numbers close to fixed points are also far more likely to occur.
The question is, can we graphically show how likely each point in the complex plane is to appear in such an iteration sequence? Assuming a random input gives essentially random outputs, it shouldn't matter what the input value is (except in the 0% chance that we pick a point that loops or eventually becomes real). I'm thinking that a logarithmic (or superlogarithmic) scaling of the probability would help bring out the details.
We wouldn't need a whole lot of precision in our math library, because once we get into the range of "large" numbers, if the exponentiation exceeds the precision of our library, we could just pick a random modulus with an appropriate distribution around 0, and a random argument, and start over.
~ Jay Daniel Fox
