uniqueness

[tommy1729]

conjecture

if f(f(x)) = exp(x)


f(x) is real -> real and > x

and f(x) is Coo

then f(x) is the unique solution to the above.

No, it is not unique. We discussed that already here. Even if you demand that f is analytic.

ok , lets see.

f(f(x)) = x^4

f(x) = x^2

another solution :

f(x) = (x + q(x)) ^2 = x^2 + 2q(x)x + q(x)^2

f(f(x)) = ( x^2 + 2q(x)x + q(x)^2 + q(x^2 + 2q(x)x + q(x)^2) ) ^2

=>

( x^2 + 2q(x)x + q(x)^2 + q(x^2 + 2q(x)x + q(x)^2) ) ^2 = x^4

=> x^2 + 2q(x)x + q(x)^2 + q(x^2 + 2q(x)x + q(x)^2) = x^2

=> 2q(x)x + q(x)^2 + q(x^2 + 2q(x)x + q(x)^2) = 0

this seems very different from your 1-periodic condition of q(x) ??