TITLE: Smooth Julia set for quadratic polynomials
QUESTION [7 upvotes]: 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.

REPLY [7 votes]: 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.
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.