DYNAMICS OF TRANSCENDENTAL MEROMORPHIC FUNCTIONS

The paper examines some properties of the dynamics of entire functions which extend to general meromorphic functions and also some properties which do not. For a transcendental meromorphic function f(z) whose Fatou set F(f) has a component of connectivity at least three, it is shown that singleton components are dense in the Julia set J(f). Some problems remain open if all components are simply or doubly connected. Let I(f) denote the set of points whose forward orbits tend to ∞ but never land at ∞. For a transcendental meromorphic function f(z) we have J(f) = ∂I(f), I(f) ∩ J(f) ̸ = ∅. However in contrast to the entire case, the components of I(f) need not be unbounded, even if f(z) has only one pole. If f(z) has finitely many poles then, as in the entire case, F(f) has at most one completely invariant component.