We investigate some connectedness properties of the set of points K(f) where the iterates of an entire function f are bounded. In particular, we describe a class of transcendental entire functions for which an analogue of the Branner-Hubbard conjecture holds and show that, for such functions, if K(f) is disconnected then it has uncountably many components. We give examples to show that K(f) can be totally disconnected, and we use quasiconformal surgery to construct a function for which K(f) has a component with empty interior that is not a singleton.
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