# On the quasiconformal equivalence of dynamical Cantor sets

## TL;DR

This paper investigates when the complements of certain Cantor sets in the complex plane, especially those generated dynamically, are quasiconformally equivalent as Riemann surfaces, focusing on Cantor Julia sets and random Cantor sets.

## Contribution

It provides new insights into the quasiconformal equivalence of Riemann surfaces formed by dynamical Cantor sets, including Julia and random Cantor sets.

## Key findings

- Quasiconformal equivalence criteria for Cantor Julia set complements
- Extension of results to random Cantor sets
- Characterization of dynamical Cantor set complements as Riemann surfaces

## Abstract

The complement of a Cantor set in the complex plane is itself regarded as a Riemann surface of infinite type. The problem is the quasiconformal equivalence of such Riemann surfaces. Particularly, we are interested in Riemann surfaces given by Cantor sets which are created through dynamical methods. We discuss the quasiconformal equivalence for the complements of Cantor Julia sets of rational functions and random Cantor sets.