Local rigidity of Schottky maps

TL;DR

This paper studies the local rigidity of Schottky maps, which are conformal maps between relative Schottky sets, contributing to the understanding of their geometric properties and implications for Julia sets of rational functions.

Contribution

It introduces the concept of Schottky maps and investigates their local rigidity, extending previous work on relative Schottky sets and aiding in the proof of quasisymmetric rigidity of Julia sets.

Findings

Schottky maps exhibit local rigidity properties.

The work advances understanding of conformal maps between Schottky sets.

Results are relevant for the study of Julia sets of rational functions.

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.