An investigation of multiplicative invariance in the complex plane

Neil MacVicar

Published 2023-01-23Version 1

Multiplicative invariance is a well-studied notion in the unit interval. The picture in the complex plane is less developed. This document introduces an analogous notion of multiplicative invariance in the complex plane and establishes similar results of Furstenberg's in this setting. Namely, that the Hausdorff and box-counting dimensions of a multiplicatively invariant subset are equal and, furthermore, are equal to the normalized topological entropy of an underlying subshift. We also extend a formula for the box-counting dimension of base-$b$ restricted digit sets where $b$ is a suitably chosen Gaussian integer.