We prove that every transcendental meromorphic map f with a disconnected
Julia set has a weakly repelling fixed point. This implies that the Julia set
of Newton's method for finding zeroes of an entire map is connected. Moreover,
extending a result of Cowen for holomorphic self-maps of the disc, we show the
existence of absorbing domains for holomorphic self-maps of hyperbolic regions
whose iterates tend to a boundary point. In particular, the results imply that
periodic Baker domains of Newton's method for entire maps are simply connected,
which solves a well-known open question.Comment: 34 pages, 10 figure
