# Is the function Point -> Julia set “injective”?

Consider the functions $f_c(z) := z^2 + c$ for $c \in \mathbb C$. For each such function, we may form the associated Julia set. My question: If $c, c' \in \mathbb C$ produce in this way the same Julia set, does this imply $c = c'$?

Trivially this is the case if we "consider one more dimension" by taking orbits into account. But if we consider the Julia set only, I can't find the solution.

The theorem says in particular that, for two centered polynomials of degrees $n,m \geq 2$ respectively, if these polynomials have the same Julia set which is not a circle or an interval, then (up to a symmetry) they are both iterates of the same polynomial. `Centered' means that there is no term of degree $n-1$, resp. $m-1$ (the corresponding coefficients are zero).