07/05/2010, 09:59 PM

more about those cycles and attractors :

you can find g that can't be written as f°f°...°f for any n > 1.

Namely, suppose g has exactly one fixed point p_0, and exactly one p_1 <> p_0 such that

g(p_1) = p_0, and at least one p_2 such that g(p_2) = p_1. If g = f°f°...°f, it's

easy to see that f(p_0) = p_0, f(p_1) = p_0, and (for some p_3) f(p_3) = p_1.

But then f(f(p_3)) = p_0 which would imply g(p_3) = p_0, contradiction.

A suitable g is g(x) = x + 2 - 2 exp(x).

hope that helps

regards

tommy1729

you can find g that can't be written as f°f°...°f for any n > 1.

Namely, suppose g has exactly one fixed point p_0, and exactly one p_1 <> p_0 such that

g(p_1) = p_0, and at least one p_2 such that g(p_2) = p_1. If g = f°f°...°f, it's

easy to see that f(p_0) = p_0, f(p_1) = p_0, and (for some p_3) f(p_3) = p_1.

But then f(f(p_3)) = p_0 which would imply g(p_3) = p_0, contradiction.

A suitable g is g(x) = x + 2 - 2 exp(x).

hope that helps

regards

tommy1729