Maybe the previous animation is even a bit misleading, not only \(\lim_{c\to 1}f^{\circ \frac{1}{2}}(x) = 0\), but rather:
\[ \lim_{c\to 1} f^{\circ t}(x) = 0,\quad {\rm for all}\quad t\notin\mathbb{Z} \]
I marked t=0.01 as a sample in this graph:
Compared to \(f'(0)=1\) where the iterates of a point remain in the same petal, for any other \(f'(0)=e^{i\varphi}\), \(0<\varphi<2\pi\) the iterates need to traverse around the fixed point 0. And it seems some get entangled there
\[ \lim_{c\to 1} f^{\circ t}(x) = 0,\quad {\rm for all}\quad t\notin\mathbb{Z} \]
I marked t=0.01 as a sample in this graph:
Compared to \(f'(0)=1\) where the iterates of a point remain in the same petal, for any other \(f'(0)=e^{i\varphi}\), \(0<\varphi<2\pi\) the iterates need to traverse around the fixed point 0. And it seems some get entangled there