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.