February  2020, 40(2): 753-766. doi: 10.3934/dcds.2020060

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

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.

Citation: Kanji Inui, Hikaru Okada, Hiroki Sumi. The Hausdorff dimension function of the family of conformal iterated function systems of generalized complex continued fractions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 753-766. doi: 10.3934/dcds.2020060
References:
[1]

C. Bandt and S. Graf, Self-similar sets. VII. A characterization of self-similar fractals with positive Hausdorff measure, Proc. Amer. Math. Soc., 114 (1992), 995-1001.  doi: 10.2307/2159618.  Google Scholar

[2]

M. F. Barnsley, Fractals Everywhere, Academic Press Professional, Boston, MA, 1993.  Google Scholar

[3]

J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.  doi: 10.1512/iumj.1981.30.30055.  Google Scholar

[4]

K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Ltd., Chichester, 1990.  Google Scholar

[5]

R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, Proceedings of the London Mathematical Society, 73 (1996), 105-154.  doi: 10.1112/plms/s3-73.1.105.  Google Scholar

[6]

R. D. Mauldin and M. Urbański, Conformal iterated function systems with applications to the geometry of continued fractions, Trans. Amer. Math. Soc., 351 (1999), 4995-5025.  doi: 10.1090/S0002-9947-99-02268-0.  Google Scholar

[7]

M. Moran, Hausdorff measure of infinitely generated self-similar sets, Monatsh. Math., 122 (1996), 387-399.  doi: 10.1007/BF01326037.  Google Scholar

[8]

M. Roy and M. Urbański, Regularity properties of Hausdorff dimension in infinite conformal iterated function systems, Ergodic Theory Dynam. Systems, 25 (2005), 1961-1983.  doi: 10.1017/S0143385705000313.  Google Scholar

[9]

A. Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc., 122 (1994), 111-115.  doi: 10.1090/S0002-9939-1994-1191872-1.  Google Scholar

[10]

R. Stankewitz, Density of repelling fixed points in the Julia set of a rational or entire semigroup, Ⅱ, Discrete Contin. Dyn. Syst., 32 (2012), 2583-2589.  doi: 10.3934/dcds.2012.32.2583.  Google Scholar

[11]

H. Sugita, Dimension of Limit Sets of IFSs of Complex Continued Fractions (in Japanese), Master thesis, under supervision of H. Sumi, Osaka University, 2014. Google Scholar

[12]

S. Takemoto, Properties of the Family of CIFSs of Generalized Complex Continued Fractions (in Japanese), Master thesis, under supervision of H. Sumi, Osaka University, 2015. Google Scholar

show all references

References:
[1]

C. Bandt and S. Graf, Self-similar sets. VII. A characterization of self-similar fractals with positive Hausdorff measure, Proc. Amer. Math. Soc., 114 (1992), 995-1001.  doi: 10.2307/2159618.  Google Scholar

[2]

M. F. Barnsley, Fractals Everywhere, Academic Press Professional, Boston, MA, 1993.  Google Scholar

[3]

J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.  doi: 10.1512/iumj.1981.30.30055.  Google Scholar

[4]

K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Ltd., Chichester, 1990.  Google Scholar

[5]

R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, Proceedings of the London Mathematical Society, 73 (1996), 105-154.  doi: 10.1112/plms/s3-73.1.105.  Google Scholar

[6]

R. D. Mauldin and M. Urbański, Conformal iterated function systems with applications to the geometry of continued fractions, Trans. Amer. Math. Soc., 351 (1999), 4995-5025.  doi: 10.1090/S0002-9947-99-02268-0.  Google Scholar

[7]

M. Moran, Hausdorff measure of infinitely generated self-similar sets, Monatsh. Math., 122 (1996), 387-399.  doi: 10.1007/BF01326037.  Google Scholar

[8]

M. Roy and M. Urbański, Regularity properties of Hausdorff dimension in infinite conformal iterated function systems, Ergodic Theory Dynam. Systems, 25 (2005), 1961-1983.  doi: 10.1017/S0143385705000313.  Google Scholar

[9]

A. Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc., 122 (1994), 111-115.  doi: 10.1090/S0002-9939-1994-1191872-1.  Google Scholar

[10]

R. Stankewitz, Density of repelling fixed points in the Julia set of a rational or entire semigroup, Ⅱ, Discrete Contin. Dyn. Syst., 32 (2012), 2583-2589.  doi: 10.3934/dcds.2012.32.2583.  Google Scholar

[11]

H. Sugita, Dimension of Limit Sets of IFSs of Complex Continued Fractions (in Japanese), Master thesis, under supervision of H. Sumi, Osaka University, 2014. Google Scholar

[12]

S. Takemoto, Properties of the Family of CIFSs of Generalized Complex Continued Fractions (in Japanese), Master thesis, under supervision of H. Sumi, Osaka University, 2015. Google Scholar

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