, Volume 96, Issue 1, pp 225-245

Escaping points of meromorphic functions with a finite number of poles

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Abstract

We establish several new properties of the escaping setI(f)={z∶f n (z)→∞ andf n(z)⇑∞ for eachn∈N} of a transcendental meromorphic functionf with a finite number of poles. By considering a subset ofI(f) where the iterates escape about as fast as possible, we show thatI(f) always contains at least one unbounded component. Also, iff has no Baker wandering domains, then the setI(f)J(f), whereJ(f) is the Julia set off, has at least one unbounded component. These results are false for transcendental meromorphic functions with infinitely many poles.