  In this paper we study the topology of a set naturally arising from the study
of $\beta$-expansions. After proving several elementary results for this set we
study the case when our base is Pisot. In this case we give necessary and
sufficient conditions for this set to be finite. This finiteness property will
allow us to generalise a theorem due to Schmidt and will provide the motivation
for sufficient conditions under which the growth rate and Hausdorff dimension
of the set of $\beta$-expansions are equal and explicitly calculable.
