Packing measure and dimension of the limit sets of IFSs of generalized complex continued fractions

TL;DR

This paper investigates the geometric properties of limit sets generated by conformal iterated function systems of generalized complex continued fractions, establishing the equality of packing and Hausdorff dimensions and finiteness of the packing measure.

Contribution

It proves the equality of packing and Hausdorff dimensions and finiteness of the packing measure for these limit sets, extending previous results on their measure properties.

Findings

Packing and Hausdorff dimensions are equal for the limit sets.

The proper-dimensional packing measure of the limit set is finite.

The Hausdorff measure of the limit set is zero, but the packing measure is positive.

Abstract

We consider a family of conformal iterated function systems (for short, CIFSs) of generalized complex continued fractions. Note that in our previous paper we showed that the proper-dimensional Hausdorff measure of the limit set is zero and the packing measure of the limit set with respect to the Hausdorff dimension is positive. In this paper, we show that the packing dimension and the Hausdorff dimension of the limit set of each CIFS in the family are equal, and the proper-dimensional packing measure of the limit set is finite.