TITLE: Is the function Point -> Julia set "injective"?
QUESTION [5 upvotes]: 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.

REPLY [6 votes]: You can deduce a positive answer to your question from Theorem 1 in the following paper:
P. Atela, J. Hu,
Commuting polynomials and polynomials with same Julia
set,
Internat. J. Bifur. Chaos Appl. Sci. Engrg. 6 (1996), no. 12A, 24
27–2432 (preprint available here: https://arxiv.org/pdf/math/9504210.pdf)
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).