So I was approaching that from a different angle:
\[ -z + z^2 = \lim_{c \to 1} -cz+z^2 \]
Because for \(0<c<1\) we have the regular iteration, which allows us to compute \(f^{\circ t}_c(z_0)\) for \(z_0 \) in the basin of attraction of 0.
Just drawing for c=0.5 and c=0.9:
So what happens if we further move \(c\to 1\):
I guess in the limit \(f^{\circ \frac{1}{2}}(x) = 0 = f^{\circ \frac{3}{2}}(x)\) for \(x>0\) (and in consequence for all real \(x\)).
And this is the reason, why the half iterate does not exist.
\[ -z + z^2 = \lim_{c \to 1} -cz+z^2 \]
Because for \(0<c<1\) we have the regular iteration, which allows us to compute \(f^{\circ t}_c(z_0)\) for \(z_0 \) in the basin of attraction of 0.
Just drawing for c=0.5 and c=0.9:
So what happens if we further move \(c\to 1\):
I guess in the limit \(f^{\circ \frac{1}{2}}(x) = 0 = f^{\circ \frac{3}{2}}(x)\) for \(x>0\) (and in consequence for all real \(x\)).
And this is the reason, why the half iterate does not exist.