# Dynamical Aspects of Piecewise Conformal Maps

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## 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.

## Keywords

Piecewise conformal maps Piecewise transformations Julia and Fatou sets Spider Web set Kleinian groups Schottky groups Limit set Structural stability## Mathematics Subject Classification

37F05 37F15 37F50 37F99## Notes

### Acknowledgements

This work was partially supported by PAPIIT IN 102515 and CONACYT CB15/255633.

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