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 and Continuous Dynamical Systems, 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.

[2]

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

[3]

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

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

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.

[2]

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

[3]

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

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[1]

Lulu Fang, Min Wu. Hausdorff dimension of certain sets arising in Engel continued fractions. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2375-2393. doi: 10.3934/dcds.2018098

[2]

Doug Hensley. Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2417-2436. doi: 10.3934/dcds.2012.32.2417

[3]

Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235

[4]

Krzysztof Barański. Hausdorff dimension of self-affine limit sets with an invariant direction. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1015-1023. doi: 10.3934/dcds.2008.21.1015

[5]

Marc Kessböhmer, Bernd O. Stratmann. On the asymptotic behaviour of the Lebesgue measure of sum-level sets for continued fractions. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2437-2451. doi: 10.3934/dcds.2012.32.2437

[6]

Laura Luzzi, Stefano Marmi. On the entropy of Japanese continued fractions. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 673-711. doi: 10.3934/dcds.2008.20.673

[7]

Pierre Arnoux, Thomas A. Schmidt. Commensurable continued fractions. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4389-4418. doi: 10.3934/dcds.2014.34.4389

[8]

David Cheban, Cristiana Mammana. Continuous dependence of attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 499-515. doi: 10.3934/dcds.2007.18.499

[9]

Tomasz Szarek, Mariusz Urbański, Anna Zdunik. Continuity of Hausdorff measure for conformal dynamical systems. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4647-4692. doi: 10.3934/dcds.2013.33.4647

[10]

Welington Cordeiro, Manfred Denker, Michiko Yuri. A note on specification for iterated function systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3475-3485. doi: 10.3934/dcdsb.2015.20.3475

[11]

Shmuel Friedland, Gunter Ochs. Hausdorff dimension, strong hyperbolicity and complex dynamics. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 405-430. doi: 10.3934/dcds.1998.4.405

[12]

Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118.

[13]

Krzysztof Barański, Michał Wardal. On the Hausdorff dimension of the Sierpiński Julia sets. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3293-3313. doi: 10.3934/dcds.2015.35.3293

[14]

Markus Böhm, Björn Schmalfuss. Bounds on the Hausdorff dimension of random attractors for infinite-dimensional random dynamical systems on fractals. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3115-3138. doi: 10.3934/dcdsb.2018303

[15]

Joseph Squillace. Estimating the fractal dimension of sets determined by nonergodic parameters. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5843-5859. doi: 10.3934/dcds.2017254

[16]

Richard Sharp. Conformal Markov systems, Patterson-Sullivan measure on limit sets and spectral triples. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2711-2727. doi: 10.3934/dcds.2016.36.2711

[17]

Michael Barnsley, James Keesling, Mrinal Kanti Roychowdhury. Special issue on fractal geometry, dynamical systems, and their applications. Discrete and Continuous Dynamical Systems - S, 2019, 12 (8) : i-i. doi: 10.3934/dcdss.201908i

[18]

Saisai Shi, Bo Tan, Qinglong Zhou. Best approximation of orbits in iterated function systems. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4085-4104. doi: 10.3934/dcds.2021029

[19]

Claudio Bonanno, Carlo Carminati, Stefano Isola, Giulio Tiozzo. Dynamics of continued fractions and kneading sequences of unimodal maps. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1313-1332. doi: 10.3934/dcds.2013.33.1313

[20]

Élise Janvresse, Benoît Rittaud, Thierry de la Rue. Dynamics of $\lambda$-continued fractions and $\beta$-shifts. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1477-1498. doi: 10.3934/dcds.2013.33.1477

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (271)
  • HTML views (99)
  • Cited by (1)

Other articles
by authors

[Back to Top]