  The Fatou-Julia theory for rational functions has been extended both to
transcendental meromorphic functions and more recently to several different
types of quasiregular mappings in higher dimensions. We extend the iterative
theory to quasimeromorphic mappings with an essential singularity at infinity
and at least one pole, constructing the Julia set for these maps. We show that
this Julia set shares many properties with those for transcendental meromorphic
functions and for quasiregular mappings of punctured space.
