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. 



A question about Julia set for quadratic family
I expect that you will find the answer (and a lot more) in the following paper, since it's easy to figure out which quadratic polynomials (if any) commute with one another:


Commuting polynomials and polynomials with same Julia set,
Pau Atela, Jun Hu,
 *Int. J. Bifurcation Chaos* **06**, 2427 (1996). DOI: <http://dx.doi.org/10.1142/S0218127496001570>


*Abstract*: It has been known since Julia that polynomials commuting under composition have the same Julia set. More recently in the works of Baker and Eremenko, Fern\'andez, and Beardon, results were given on the converse question: When do two polynomials have the same Julia set? We give a complete answer to this question and show the exact relation between the two problems of polynomials with the same Julia set and commuting pairs. 


The paper is also available on the ArXiv


<https://arxiv.org/abs/math/9504210>


