Existence of self-similar embedded fractals

Let a fractal be defined by

$$z \leftarrow f(z)+c.$$

For a positive scaling factor $s$, a complex translation point $a=(x_n,y_n)$, and a rotation angle $\phi$, define the zoom transform by

$$z \leftarrow s e^{i\phi}f(z-a)+c.$$

Existence of self-similar embedded fractals. A fractal of the form $z \leftarrow f(z)+c$ has quasi-self-similar embedded fractals under magnification if and only if it is invariant under the zoom transform.