Abel function

Remarks

Example: for $f(x) = 3x$ on $X = (0, \infty)$, $\alpha(x) = \log_3 x$ is an Abel function, since $\log_3(3x) = \log_3 x + 1$. Non-example: for the same $f$, $\alpha(x) = \ln x$ is not an Abel function, since $\ln(3x) = \ln x + \ln 3$ and $\ln 3 \ne 1$. If $\alpha$ is an injective Abel function and $t$ is real with $\alpha(X) + t \subseteq \alpha(X)$, then $f^t(x) = \alpha^{-1}(\alpha(x) + t)$ satisfies $f^1 = f$ and $f^s \circ f^t = f^{s + t}$, so $(x, t) \mapsto f^t(x)$ solves the translation equation where it is defined. In the example, $f^t(x) = 3^t x$.