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Suppose a sequence of analytic maps $f_n: \mathbb{D} \to \mathbb{D}$ from the unit disk to itself, each of which is a topological covering map to its image, converges locally uniformly to an analytic map $f$. Under what conditions will the limit $f$ also be a topological covering map to its image? (I am also interested in other possibilites for domain and range of the $f_n$.)

Using Rouche's theorem, one can show that the local uniform limit of injective analytic functions is again injective, provided that the limit is not constant. One thing I would like to know is if a similar result holds if we replace "injective" with "covering map".

This sort of question comes up if you try to prove the uniformization theorem in the special case of a plane domain $U\subset \mathbb{C}$.

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Suppose a sequence of analytic covering maps $f_n: \mathbb{C} mathbb{D} \to \mathbb{D}$ from the unit disk to itself, each of which is a covering map to its image, converges locally uniformly to an analytic map $f$. Under what conditions will the limit $f$ also be a covering map to its image? (I am also interested in other possibilites for domain and range of the $f_n$.)

Using Rouche's theorem, one can show that the local uniform limit of injective analytic functions is again injective, provided that the limit is not constant. One thing I would like to know is if a similar result holds if we replace "injective" with "covering map".

This sort of question comes up if you try to prove the uniformization theorem in the special case of a plane domain $U\subset \mathbb{C}$.

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