This question is related to a classification of rational maps in terms of properties of their Julia set. 


Let $f= z^2 + c$, with $c\in \mathbb{C}$ be a quadratic polynomial such that its Julia set $J(f)$ is connected. 


* Q1: If there exists a relatively open set of $J(f)$ that is (the support of) a smooth curve. Is $f$ conjugate (resp equal) to a Tchebychev polynomial or power map $z^2$?
* Q2: Is the answer to Q1 yes under the additional assumption that $J(f)$ is also locally connected?
* Q3: If the answer to Q1 is no, can one describe the set of such counterexamples in terms of the parameter $c$?


Thanks a lot. 



Smooth Julia set for quadratic polynomials
The answer to a) is yes, and this was proved by Fatou in 1919. 
[Sur les équations fonctionnelles
Bulletin de la S. M. F., tome 48 (1920), p. 208-314](http://archive.numdam.org/article/BSMF_1920__48__208_1.pdf).
There are many generalizations of this fact. For one generalization, and further references you may look to
[Meromorphic functions with linearly distributed values and Julia sets of rational functions, Proc. AMS. 137 (2009), 2329-2333.](http://www.math.purdue.edu/~eremenko/dvi/real-2.pdf)


