Daniel Wrote: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. I would love to see references to published material.
I think this is the oldest reference to continuously iterating functions:
G. Koenigs, Recherches sur les intégrales de certaines équations fonctionnelles, Annales sci. de l'École Normale Supérieure (3) 1 (1884), Supp. 3-41.
Koenigs showed in 1884 that if we have a power series
\( f(z)=\sum_{n=1}^\infty a_n z^n \) with \( 0<|a_1|<1 \) which is convergent for \( |z|<R_1 \) then the function
\( \chi(z)=\lim_{n\to\infty} a_1^{-n} f^{\circ n}(z) \) exists and is analytic in a suitable neighborhood of 0 and is a solution of the Schroeder equation \( \chi(f(z))=a_1 \chi(z) \).
Given this we easily derive continuous iterates by
\( f^{\circ t}(z)=\chi^{-1}(a_1^t \chi(z)) \).
So i dont know what you mean by a "continuously iterated function that displayed chaotic behavior" but surely methods for continuously iterating analytic functions with hyperbolic fixed point were known already around 1884.
