Continuous iteration

[jaydfox]

Cris Moore asked about the compatability of solutions from different fixed points. By using a fractal with low entropy I was able to experimentally show the correct logrithmic spiral of a neighboring fixed point ...

So does that mean the solutions are equal? Id rather guessed that they are different for different fixed points. Did you compute the fixed points and their derivative of \( e^x \)? Are they all attracting?

I think the first fixed point I used was attracting and the neighoring fixed point was repelling. In general the hyperbolic fixed points are repelling and are found using iterated logs with a given branch.

Numerically the difficultly is that the dynamics are simple at the two fixed points but are complicated in the chatotic area between them. I used something like \( 1.1^x \) so as to mimimize the chaotic areas of the Julia set between two fixed points. This doesn't prove that the solutions are equal, it just provided numerial evidence that they are equal.

I stumbled upon this while studying fixed points for bases less than 1. I didn't realize that I was observing the same fixed points as you.

If we take the natural logarithm of the real interval (0, 1), we get a curve with endpoints \( (-\infty, 0) \), and the next iteration has endpoints \( (+\infty+\pi i, -\infty + \pi i) \). Notice that the modulus of each endpoint is infinity. Therefore, all iterates thereafter will have endpoints with positive infinity for the real part, and (assuming the principal branch) imaginary part between 0 and pi.

Each successive iteration of the natural logarithm of the real interval (0,1) will drill deeper towards the fixed point, but because the endpoints are at infinity, each successive iterate will have to snake its way between outer iterates to get out. The curves are non-intersecting.

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.

Continuous iteration would therefore have to stick to real values for real iteration counts (for the unit interval), to avoid these singularities. I don't see another way to avoid the singularities. This means that continuous iteration near the fixed point would have to follow a particular curve. For those reading this, you'd have to see what these curves look like to get my meaning here.

I've only been studying base e, so it may be possible for other bases greater than eta to avoid these singularities, but I kind of doubt it.