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I am reading the paper by Douady and Hubbard, On the Dynamics of Polynomial-Like Mappings, and I am at a loss to understand a crucial step in the construction of the right domain for finding the external map to a polynomial-like mapping.

Do you know a reference where the construction of the repeated covering by annuli (in the case $K_f$ is disconnected) is both justified and done in detail?

I have too many doubts to ask a specific closed question about it. Thank you very much.

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    $\begingroup$ Without more detail (ideally without having to refer back to the paper in question, if possible) it is essentially impossible to answer you. $\endgroup$ – Lasse Rempe Mar 25 '16 at 18:30
  • $\begingroup$ @LasseRempe-Gillen Thanks for the comment. Somehow I thought this proof was common knowledge as its part of the proof for the Straightening theorem and this is a seminal paper. I will think about rewriting a more specific question. $\endgroup$ – Andrea Mar 25 '16 at 19:41
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    $\begingroup$ Of course the proof of the Straightening Theorem is well-known. But it doesn't help if you don't even specify what "this proof" is that you are talking about ... Also keep in mind that different people may have learned from different sources, and possibly a long time ago (e.g. I learned from both Lyubich and Douady about the topic, and they had somewhat different perspectives). A good general rule is: If you would like people to take the time to answer your question, make it as easy for them as possible ... $\endgroup$ – Lasse Rempe Mar 25 '16 at 22:17