  For a transcendental entire function f, we study the set of points BU(f)
whose iterates under f neither escape to infinity nor are bounded. We give new
results on the connectedness properties of this set and show that, if U is a
Fatou component that meets BU(f), then most boundary points of U (in the sense
of harmonic measure) lie in BU(f). We prove this using a new result concerning
the set of limit points of the iterates of f on the boundary of a wandering
domain. Finally, we give some examples to illustrate our results.
