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Pure Mathematics 2020
广义复连分数共形迭代系统的Hausdorff维数
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Abstract:
本文研究了含有复参数的一族广义复连分数共形迭代系统。Sumi等利用无限生成共形迭代系统理论研究了广义复连分数，得到了关于广义复连分数共形迭代系统极限集的Hausdorff维数的一系列结果。本文进一步将Sumi等研究的共形迭代系统的参数推广到更大的区域，对于这个具有更大参数空间的广义连分数共形迭代系统，证明了其极限集的Hausdorff维数在参数空间上是连续的，在参数空间内部是连续的且实解析和次调和的。并由此得到Hausdorff维数在参数空间的边界点上取到最大值。
In this article, we consider a family of conformal iterated function systems (CIFSs) of generalized complex continued fractions with a complex parameter in a domain. Sumi et al. studied the general complex continued fractions by applying the theory of CIFSs generated by infinite many conformal maps, and got a series of interesting results. We further generalize the CIFS studied by Sumi et al. to a larger parameter domain. We prove that the Hausdorff dimension function of the limit sets of CIFSs of generalized complex continued fraction is continuous in the parameter domain and is real-analytic and subharmonic in the interior of the parameter domain. As a consequence, the Hausdorff dimension function assumes maximum value on the boundary of the parameter domain.
| [1] | Mauldin, R.D. and Urbanski, M. (1996) Dimensions and Measures in Infinite Iterated Function Systems. Proceedings of the London Mathematical Society, 73, 105-154. https://doi.org/10.1112/plms/s3-73.1.105 |
| [2] | Mauldin, R.D. and Urbanski, M. (1999) Conformal Iterated Function Systems with Applications to the Geometry of Continued Fractions. Transactions of the American Mathematical Society, 351, 4995-5025. https://doi.org/10.1090/S0002-9947-99-02268-0 |
| [3] | Inui, K., Sumi, H. and Okada, H. (2020) The Hausdorff Dimension Function of the Family of Conformal Iterated Function Systems of Generalized Complex Continued Fractions. Discrete and Continuous Dynamical Systems, 40, 753-766. https://doi.org/10.3934/dcds.2020060 |
| [4] | Inui, K. and Sumi, H. (2020) Hausdorff Measures and Packing Measures of Limit Sets of CIFSs of Generalized Complex Continued Fractions. Journal of Difference Equations & Applications, 26, 104-121. https://doi.org/10.1080/10236198.2019.1709063 |
| [5] | Roy, M. and Urbanski, M. (2005) Regularity Properties of Hausdorff Dimension in Infinite Conformal Iterated Function Systems. Ergodic Theory and Dynamical Systems, 25, 1961-1983. https://doi.org/10.1017/S0143385705000313 |
| [6] | Qiu, W.Y., Wang, Y.F., Yang, J.H. and Yin, Y.C. (2014) On Metric Properties of Limit Sets of Contractive Analytic Non-Archimedean Dynamical Systems. Journal of Mathematical Analysis and Applications, 414, 386-401. https://doi.org/10.1016/j.jmaa.2014.01.015 |
| [7] | 邱维元. 一类Julia集的Hausdorff维数的上界估计[J]. 复旦学报自然科学版, 1993(2): 44-50. |
| [8] | Bandt, C. and Graf, S. (1992) Self-Similar Sets VIIA Characterization of Self-Similar Fractals with Positive Hausdorff Measure. Proceedings of the American Mathematical Society, 114, 995-1001. https://doi.org/10.2307/2159618 |
| [9] | Hutchinson, J. (1981) Fractals and Self-Similarity. Indiana Uni-versity Mathematics Journal, 30, 713-747. https://doi.org/10.1512/iumj.1981.30.30055 |
| [10] | Falconer, K. (1990) Fractals Geometry: Mathematical Foundations and Applications. Biometrics, 46, 886-887. https://doi.org/10.2307/2532125 |
| [11] | Moran, M. (1996) Hausdorff Measure of Infinitely Generated Self-Similar Sets. Monatshefte für Mathematik, 122, 387-399. https://doi.org/10.1007/BF01326037 |
| [12] | Fan, A.H., Liao, L.M., Wang, Y.F. and Zhou, D. (2007) P-adic Repellers in Qp Are Subshifts of Finite Type. Comptes Rendus Mathematique, 344, 219-224. https://doi.org/10.1016/j.crma.2006.12.007 |
| [13] | Schief, A. (1994) Separation Properties for Self-Similar Sets. Proceedings of the American Mathematical Society, 122, 111-115. https://doi.org/10.1090/S0002-9939-1994-1191872-1 |
