fractional iterate of x ?

[bo198214]

Any function with symmetric against y=x line plot is half-iterate of x.

Yes, these functions are called involutions. And they are roughly the only solutions \( \varphi \) of the Babbage equation
(*) \( \varphi^{[N]}=\operatorname{id} \).

Theorem 11.7.1 in "Iterative functional equations":
Let a self-mapping \( \varphi \) of a real interval be a continuous solution of equation (*). Then either \( \varphi \) itself is the identity mapping or \( N \) has to be even and \( \varphi \) is a strictly decreasing involution.

I like to compare this with real numbers:
"Let a real number x be a solution of \( x^N=1 \). Then either \( x \) is 1 itself or \( N \) has to be even and \( x \) is a negative number."