Weakly repelling fixpoints and the connectivity of wandering domains

Abstract

It is proved that if a transcendental meromorphic function f f has a multiply-connected wandering domain, then f f has a fixpoint z 0 z_0 such that | f ′ ( z 0 ) | > 1 |f’(z_0)|>1 or f ′ ( z 0 ) = 1 f’(z_0)=1 . Entire functions with a multiply-connected wandering domain have infinitely many such fixpoints. These results are used to show that solutions of certain differential equations do not have wandering domains at all.