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

Let $f= z^2 + c$, with $cin mathbbC$ 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.

From this paper of Bedford and Kim (arxiv link):

Fatou showed that if the
Julia set $J$ is a smooth curve, then either $J$ is the unit circle, or $J$ is a real interval. If $J$ is the circle, then $f$ is equivalent to $z → z^d$
, where $d$ is an integer with $|d| ≥ 2$; if $J$ is the interval, then $f$ is equivalent to a Chebyshev polynomial.

The answer to a) is yes, and this was proved by Fatou in 1919. 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.