Finite Hausdorff measure conjecture for large-parameter complex infinite IFSs

Let $\mathcal{F}$ and $\mathcal{G}$ be the two CIFSs considered in the paper, with limit sets $J_{\mathcal{F}}$ and $J_{\mathcal{G}}$, Hausdorff dimensions $h_{\mathcal{F}}$ and $h_{\mathcal{G}}$. Let $T$, $c_2$, $\gamma$, and $N$ be the parameters appearing in the construction.

Finite Hausdorff measure conjecture. If

$$T\geq c_2^{2/\gamma}\cdot(N+1),$$

then

$$0<H_{h_{\mathcal{F}}}(J_{\mathcal{F}})<\infty, \qquad 0<H_{h_{\mathcal{G}}}(J_{\mathcal{G}})<\infty.$$

The preceding theorem proves only the corresponding dimension and infinite packing-measure conclusions under this condition; the asserted positivity and finiteness of the Hausdorff measures remain conjectural.