# Dynamical aspects of piecewise conformal maps

## TL;DR

This paper investigates the dynamics of piecewise conformal maps on the Riemann sphere, exploring their normality, chaos, and stability, and relating these properties to Kleinian groups and rational map iterations.

## Contribution

It introduces a framework for analyzing the dynamics of piecewise conformal maps and connects their stability to Kleinian group properties under specific conditions.

## Key findings

- Normal and chaotic regions are characterized.
- Stability is linked to Kleinian group properties.
- Several properties of these dynamical sets are established.

## Abstract

We study the dynamics of piecewise conformal maps in the Riemann sphere. The normality and chaotic regions are defined and we state several results and properties of these sets. We show that the stability of these piecewise maps is related to the Kleinian group generated by their transformations under certain hypotheses. The general motivation of the article is to compare the dynamics of piecewise conformal maps and those of the Kleinian groups and iterations of rational maps.