  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.
