Abstract
We introduce Schottky maps-conformal maps between relative Schottky sets, and study their local rigidity properties. This continues the investigations of relative Schottky sets initiated in [S. Merenkov, Planar relative Schottky sets and quasisymmetric maps, Proc. London Math. Soc. (3) 104 (2012), 455-485]. Besides being of independent interest, the latter and current works provide key ingredients in the forthcoming proof of quasisymmetric rigidity of Sierpi\'nski carpet Julia sets of rational functions.
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