Continuous iteration

[Daniel]

I spoke with Stephen Wolfram in 1986 who assured me that no solution for a continuously iterated function that displayed chaotic behavior was known at the time.

What do you mean by "chaotic behaviour"? In our case the functions to be iterated are \( b^x \), which are a rather behaving function afaik.

Requiring a fixed point is not much of a requirement. Sure the fixed points may be complex and lead to odd looking solutions, but so what?

Not exactly odd looking but simply complex for real arguments.
Thats simply not what we want All the basic functions you learn in analysis yield real values for real arguments.

To quote thr Rolling Stones, "you can't always get what you want". I didn't care for the results myself when I obtained them. Tetration is not like the functions I studied in analysis either. I now believe that these odd solutions are in fact correct, even though they take real values for integerial tetration and are complex valued otherwise.

As to fixed points being a problem, the fixed points determine the dynamics of maps based on the type of fixed point.

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.