  We introduce a family of dynamical systems that generate negative
$\beta$-expansions and study the support of the invariant measure which is
absolutely continuous with respect to Lebesgue measure. We give a
characterization of the set of digit sequences that is produced by a typical
member of this family of transformations. We discuss the meaning of greedy
expansions in the negative sense, and show that there is no transformation in
the introduced family of dynamical systems that generates negative greedy.
However, if one looks at random algorithms, then it is possible to define a
greedy expansion in base $-\beta$.
