The Limit Sets of Linear and Nonlinear Infinite IFSs Related to Complex Continued Fractions

We introduce two families of infinite iterated function systems (IFSs) $\mathcal{F}(\mathbf{d}, T)$ and $\mathcal{G}(\mathbf{d}, T)$, parametrized by a sequence of positive real numbers $\mathbf{d}$ and a natural number $T$, and investigate the measure-theoretic properties of their limit sets. $\mathcal{G}(\mathbf{d}, T)$ is an infinite M\"{o}bius IFS, which is an extension of the IFS of real continued fractions to the IFS on the closed unit disc in the complex plane. $\mathcal{F}(\mathbf{d}, T)$ is an infinite linear IFS that shares the same first approximation as $\mathcal{G}(\mathbf{d}, T)$. We show that for many choices of $\mathbf{d}$ and $T$, the limit sets of both $\mathcal{F}(\mathbf{d}, T)$ and $\mathcal{G}(\mathbf{d}, T)$ exhibit phenomena unique to infinite IFSs, such as having zero Hausdorff measure at the Hausdorff dimension, having infinite packing measure at the packing dimension, or having different Hausdorff and packing dimensions. We also prove that the Hausdorff dimension of the limit set of $\mathcal{F}(\mathbf{d}, T)$ is strictly larger than that of $\mathcal{G}(\mathbf{d}, T)$ under certain conditions on the parameters.