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

We consider a family of conformal iterated function systems (for short, CIFSs) of generalized complex continued fractions which is a generalization of the CIFS of complex continued fractions. We show the packing dimension and the Hausdorff dimension of the limit set of each CIFS in the family are equal and the packing measure of the limit set with respect to the packing dimension of the limit set is finite in order to present new and interesting examples of infinite CIFSs. Note that the Hausdorff measure of the limit set with respect to the Hausdorff dimension is zero. To prove the above results, we consider three cases (essentially two cases) and define a `nice' subset of the index set of the CIFS in each case. In addition, we estimate the cardinality of the `nice' subsets and the conformal measure of the CIFSs.Comment: arXiv admin note: text overlap with arXiv:1812.0799