Let $\gamma$ be a Jordan analytic curve on the Riemann sphere, and $f$ a rational function of degree at least 2 which maps $\gamma$ onto itself homeomorphically. The following examples of such situation are known:
a) $\gamma$ is a circle
b) $\gamma$ is a level curve of the linearizing map $\phi$ in a rotation domain. This means that there is a simply or doubly connected region where $f$ is conjugate to an irrational rotation of a disc or a ring: $$\phi\circ f=\lambda\phi,$$ where $\lambda=\exp(\pi i\alpha)$, $\alpha$ irrational.
Are there other examples? Can a rational function $f$ of degree at least $2$ map an invariant Jordan analytic curve, which intersects the Julia set $J(f)$, and which is not a circle, homeomorphically?
If one drops the condition that $f:\gamma\to\gamma$ is a homeomorphism, there are examples in http://arxiv.org/abs/1110.6552 and in the answer of Peter Mueller to my previous question Circles and rational functions.