# Computable geometric complex analysis and complex dynamics

## TL;DR

This paper explores the computability and complexity of conformal mappings, Julia sets, and related structures in complex dynamics, highlighting current understanding and challenges in the field.

## Contribution

It reviews the state of the art in computability and complexity for conformal mappings and Julia sets, providing a comprehensive overview of recent advances.

## Key findings

- Conformal mappings' computability depends on boundary regularity.
- Julia sets' computability varies with their dynamical properties.
- Complexity results for invariant measures and external rays are summarized.

## Abstract

We discuss computability and computational complexity of conformal mappings and their boundary extensions. As applications, we review the state of the art regarding computability and complexity of Julia sets, their invariant measures and external rays impressions.