Multiplicative Invariance for a Class of Subsets of the Complex Plane

TL;DR

This paper extends the concept of multiplicative invariance from the unit interval to the complex plane, establishing dimension equalities and entropy relations for such sets, and analyzing digit set intersections with Gaussian integer bases.

Contribution

It introduces a new definition of multiplicative invariance in the complex plane and proves analogous dimension and entropy results as in the real case, including for attractors of iterated function systems.

Findings

Hausdorff and box-counting dimensions are equal for multiplicatively invariant sets.

Dimensions match the normalized topological entropy of an associated subshift.

Results extend to intersections of digit sets with their translates in Gaussian integer bases.

Abstract

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.