广义复连分数共形迭代系统的Hausdorff维数

Hausdorff Dimensions of Conformal Iterated Function Systems of Generalized Complex Continued Fractions

摘要: 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。

Abstract: 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.