03/22/2013, 08:19 PM

What is the analytic solution to f(f(x))=x^2-x+1? I am thinking about Ecalle's method . . .

I don't think transforming this into Abel's equation would be of any use since it's mostly applicable where f has a attractive fixed points and transforming it into Schroeder's form and solving it by following Koening's line for 0 < |f'(z)| < 1. But x^2 - x + 1 has only one fixed point which is neutral.

I think Taylor series maybe used since 1 is probably a fixed point of f but it doesn't seems like it would converge, would it? Can we analytically continue the taylor series then?

Balarka

.

I don't think transforming this into Abel's equation would be of any use since it's mostly applicable where f has a attractive fixed points and transforming it into Schroeder's form and solving it by following Koening's line for 0 < |f'(z)| < 1. But x^2 - x + 1 has only one fixed point which is neutral.

I think Taylor series maybe used since 1 is probably a fixed point of f but it doesn't seems like it would converge, would it? Can we analytically continue the taylor series then?

Balarka

.