If we have two functions $f(x)$ and $g(x)$ I can construct a function $s_t(x)$ such that $s_0(x)=f(x)$ and $s_1(x)=g(x)$. Using iteration using:
$h(x) = f^{-1}(g(x))$
$s_t(x) = f( h^t(x) )$
Where $h^t$ is the fractional iteration of h. e.g. $h^3(x)=h(h(h(x)))$ and assuming it can be well defined for fractional values of $t$. So here $h^0(x)=x$ and $h^1(x)=f^{-1}(g(x))$.
My question is, does this function $s_t(x)$ have a name? And what are its uses?