To motivate this question, let me begin with a picture:
Each letter labels a "blob" of this quartic julia set. (is there a technical term for these parts?). Because of resolution limitations I haven't been able to color and label every blob.
The only answer to the the MO question Symmetries of Julia sets for $z^2 - c$ mentions that “This means that there is a quasi-conformal map (thus of bounded distortion) which maps parts of the Julia set to the whole.” The transformation that this labelling seems to suggest maps the entire Julia set to itself.
Does the set of quasiconformal transformations $\Xi$ (as motivated by the one above) which map the Julia set to itself form a group? I am aware that the quasiconformal transformation is not a quasiconformal conjugacy -- while it fixes two points in the plane it also fixes the entire shape of the Julia set)
If it exists, Is $\Xi$ the discrete subgroup of a continuous group? (I want to make a movie of such a transformation as this)
What relation, if any, exists between $\Xi$ and Teichmüller space?
The reason that automorphism is in quotes in the question title is because the phrase "quasiconformal automorphism groups" was used in this paper
EDIT: Here are two more of these transformations, applied to the Douady Rabbit:
The first one simply rotates around a vertex:
In the second, the point that is fixed is inside the mauve Fatou component.