Abstract:

We will discuss the existence of absorbing domains for meromorphic transcendental functions and we will show how to use them to prove the following result: Let $f$ be a meromorphic map. If the Julia set is not connected then $f$ must have at least one weakly repelling fixed point. This result is the transcendental version of a Theorem due to Shishikura for rational maps: Let $f$ be a rational map of degree at least two. If the Julia set is not connected then $f$ must have at least two weakly repelling fixed point. The results we will discuss are part of a joint work with K. Baranski and B. Karpinska.