Multiplicative Invariance for a Class of Subsets of the Complex Plane

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.