TITLE: A question about Julia set for quadratic family
QUESTION [7 upvotes]: Let $P_{c}(z)=z^2+c$. It seems from the software that the map between the parameter $c$ and the Julia set $J(P_c)$ is an injective map. Is there some reference about it?  Any comments and reference will be appreciated.

REPLY [4 votes]: For $z^2+c$ it is really easy: the answer is yes. In general, if Julia sets of polynomials coincide, these polynomials are equal unless their common Julia set has a rotation symmetry, or polynomials are permutable. 
(Baker, Eremenko, Ann. Acad. Sci. Fenn., 12 (1987) 229-236.)
http://www.acadsci.fi/mathematica/Vol12/vol12pp229-236.pdf
For $z^2+c$ we never have rotational symmetry and never have permutable polynomials. The result cited above was reproved and generalized by many other people. For rational functions, the question in full generality is still open.