Fragments of dynamic of Möebious mappings and some applications. Part I

In this article we prove that a continuous mapping on a simply-connected domain of the extended complex plane, which is normal with respect to the cycle group of all conformal automorphisms of the domain with a fixed attractive point, which belongs to the domain is a constant function. Applying this result we obtain new proofs of the classical Theorem of Liouville and little Picard Theorem for holomorphic, meromorphic and harmonic functions in complex plane. We also prove some results from the dynamic of Möbius mappings.