Local rigidity of Schottky maps

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.