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

On universal and periodic $\beta$-expansions, and the Hausdorff dimension of the set of all expansions