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Hausdorff measures and packing measures of the limit sets of CIFSs of generalized complex continued fractions

We consider a 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 dimen/sion 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.

preprint2020arXivOpen access
Kanji InuiHiroki Sumi
math.CVmath.DS
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Kanji InuiHiroki Sumi

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