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Hausdorff measures and packing measures of limit sets of CIFSs of generalized complex continued fractions |
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| Authors: | Kanji Inui Hiroki Sumi |
| Affiliation: | 1. Course of Mathematical Science, Department of Human Coexistence, Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, Japaninui.kanji.43a@st.kyoto-u.ac.jp;3. Course of Mathematical Science, Department of Human Coexistence, Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, Japan |
| Abstract: | ABSTRACTWe consider a certain family of CIFSs of the generalized complex continued fractions with a complex parameter space. We show that for each CIFS of the family, the Hausdorff measure of the limit set of the CIFS with respect to the Hausdorff dimension is zero and the packing measure of the limit set of the CIFS with respect to the Hausdorff dimension is positive (main result). This is a new phenomenon of infinite CIFSs which cannot hold in finite CIFSs. We prove the main result by showing some estimates for the unique conformal measure of each CIFS of the family and by using some geometric observations. |
| Keywords: | Infinite conformal iterated function systems fractal geometry Hausdorff measures packing measures generalized complex continued fractions |