  Multiplicative invariance is a well-studied property of subsets of the unit
interval. The theory in the complex plane is less developed. This paper
introduces an analogous definition for multiplicative invariance in the complex
plane coinciding with a more general definition concerning subsets of
attractors of iterated function systems satisfying the strong separation
condition. We establish similar results to those of Furstenberg's in the unit
interval. Namely, that the Hausdorff and box-counting dimensions of a
multiplicatively invariant set are equal and, furthermore, are equal to the
normalized topological entropy of an underlying subshift. We also extend
results concerning the box-counting dimension of intersections of base-$b$
restricted digit sets with their translates where $b$ is a suitably chosen
Gaussian integer.
