Continuous iteration

[jaydfox]

Therefore there is no direct path between iterates that wouldn't cut through an infinite number of such curves. There would be singularities where the nested curves bunch up. Therefore, the chaos will completely overwhelm any attempt at a solution that tries to iterate naively in this manner.

Hmm, I think I may need to retract that claim about the singularities. While we would have to cut through an infinite number of deeper iterates, they wouldn't necessarily be far off the mark. The nesting behavior has to do with logarithms of numbers with very large real parts and small non-zero imaginary parts, where the logarithm would end up with a moderate-sized real part and an imaginary part that is extremely close to zero. The deeper into the nesting we go, the closer to zero these imaginary parts become, so they should act convergently towards real numbers. Therefore, it's not the nested iterates of the logarithm of real numbers we need to worry about, it's all the space in between. If those regions behave nicely as well, then perhaps the chaos will not be a problem. The question remains for me: what does the solution look like, then?

I'll try to have an answer in the next few days, depending on my work schedule.