Fractional iteration

Remarks

Example: on $X = (0, \infty)$, let $f(x) = \frac{x}{1 + x}$ and $f^t(x) = \frac{x}{1 + tx}$. Clauses 1 and 2 hold by substitution, and clause 3 holds since $f^s(f^t(x)) = \frac{x/(1+tx)}{1 + sx/(1+tx)} = \frac{x}{1 + (s+t)x}$.

Non-example: on $X = \R$, the family $f^t(x) = (1 + t) x$ satisfies clauses 1 and 2 for $f(x) = 2x$, but fails clause 3, since $f^1(f^1(x)) = 4x \ne 3x = f^2(x)$.

Clause 3 with integer $s, t$ forces $f^n$ to be the usual iterate, and $f^{1/n}$ is an iterative root of order $n$. In terms of $F(x, t) = f^t(x)$, clause 3 is the translation equation $F(F(x, t), s) = F(x, s + t)$. If $\Psi \colon X \to \R$ is a bijective solution of Schröder's equation $\Psi \circ f = \lambda \Psi$ with $\lambda > 0$, then $f^t = \Psi^{-1} \circ (\lambda^t \Psi)$ is a fractional iteration. If $A \colon X \to \R$ is a bijective solution of the Abel equation $A \circ f = A + 1$, then $f^t = A^{-1} \circ (A + t)$ is one. Some authors also call a family indexed by all real $t$, or only by rational $t$, a fractional iteration.