The Hausdorff dimension function of the family of conformal iterated function systems of generalized complex continued fractions

Kanji Inui; Hikaru Okada; Hiroki Sumi

doi:10.3934/dcds.2020060

Article Contents

2020, Volume 40, Issue 2: 753-766. Doi: 10.3934/dcds.2020060

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The Hausdorff dimension function of the family of conformal iterated function systems of generalized complex continued fractions

1.
Course of Mathematical Science, Department of Human Coexistence, Graduate School of Human and Environmental Studies, Kyoto University, Yoshida-nihonmatsu-cho, Sakyo-ku, Kyoto, 606-8501, Japan
2.
Department of Mathematics, Graduate School of Science, Osaka University, 1-1, Machikaneyama-cho, Toyonaka-shi, Osaka, 560-0043, Japan

Received: October 2018

Published: November 2019

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Abstract

We consider the family of CIFSs of generalized complex continued fractions with a complex parameter space. This is a new interesting example to which we can apply a general theory of infinite CIFSs and analytic families of infinite CIFSs. We show that the Hausdorff dimension function of the family of the CIFSs of generalized complex continued fractions is continuous in the parameter space and is real-analytic and subharmonic in the interior of the parameter space. As a corollary of these results, we also show that the Hausdorff dimension function has a maximum point and the maximum point belongs to the boundary of the parameter space.

Keywords:

Infinite conformal iterated function systems,
fractal geometry,
limit sets,
Hausdorff dimension,
generalized complex continued fractions.

Mathematics Subject Classification: 28A80, 37F35.

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References

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