Hausdorff and Packing Measures for $β$-L{ü}roth Infinite Iterated Function Systems
We study the Hausdorff and packing measures of attractors generated by infinite $β$-Lüroth iterated function systems. Let $I\subset\mathbb N$ be infinite and let $\dim_H E_I=h$. For a regular system, we first give conditions for the positivity of $\mathcal H^h(E_I)$ and the finiteness of $\mathcal P^h(E_I)$ in terms of first-level Lüroth intervals. We then introduce tail-regular partitions and the scale $R_n=\frac{B_n^\infty}{B_n}.$ Under tail-regularity, the Hausdorff and packing conditions reduce to counting conditions for $I$ on intervals of length comparable to $R_n$, together with a tail condition in the packing case. Finally, we apply these results to regularly varying partitions and stretched-exponential partitions.
math.DS↗