Classification of Newton maps: realization

On the way to realize combinatorial models as real Newton maps the plan was to use Thurston’s theorem. In order to do so, one needs to prove that the branched covering, specified by the combinatorial data, doesn’t have Thurston obstructions.
This is the point where theorem of K.Pilgrim and T.Lei comes in use. It basically says that obstructions don’t intersect the preimages of arc systems.
The main difficulty in our case was to eliminate the case when the obstruction does intersect the arc system itself. Due to the theorem mentioned above this would imply that it cannot intersect the preimages of the arc systems – that is exactly the point where we aim at a contradiction.
Suppose it happens. Then as follows from the theorem the obstruction must be a Levy cycle. It is also known (BFH) that the complement components of the curves in the Levy cycles can only contain periodic mark points which in our case would be periodic points on extended Hubbard trees lying in the Julia set.
Such points being repelling for corresponding polynomial-like maps, have at least one external ray landing on it, i.e. there exists a preimage of the arc system which intersects the Thurston obstruction.


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