  Expansions in noninteger positive bases have been intensively investigated
since the pioneering works of R\'enyi (1957) and Parry (1960). The discovery of
surprising unique expansions in certain noninteger bases by Erd\H os, Horv\'ath
and Jo\'o (1991) was followed by many studies aiming to clarify the topological
and combinatorial nature of the sets of these bases. In the present work we
extend some of these studies to more general, negative or complex bases.
