The Mandelbrot Set is generated by iterating $f(z)=z^2+c$.
Is there a "half-step" function $g(z)$ such that $g(g(z))=f(z)$? What is the method for finding such half-iteration functions?
Likewise, is there a "half-half-step" (or "quarter-step") function $h(z)$ such that
$$h(h(z))=g(z)$$
and
$$h(h(h(h(z))))=f(z)$$
?
If so can we subdivide the steps to infinity? i.e.
$$g_n(g_n(z))=g_{n-1}(z)$$
and
$$g_0(z)=f(z)$$
?
And could we thereby define a continuation of the Mandelbrot iteration function?