Regularity properties of Hausdorff dimension in infinite conformal iterated function systems
Regularity properties of Hausdorff dimension in infinite conformal iterated function systems
复制标题
无限共形迭代函数系统中豪斯多夫维数的正则性质
DOI：
10.1017/s0143385705000313
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发表时间：
2005
影响因子：
0.9
通讯作者：
M. Urbanski
中科院分区：
文献类型：
--
作者：
Mario Roy;M. Urbanski
本文讨论共形迭代函数系统 (CIFS) 系列。所有 CIFS 的空间，具有共同的种子空间 X 和字母表 I，相继被赋予逐点收敛的拓扑和一种新的、较弱的拓扑，称为 $\lambda$-拓扑。证明了当 I 有限时，压力和极限集的 Hausdorff 维数相对于点收敛拓扑是连续的；当 I 无限时，压力和极限集的 Hausdorff 维数是下半连续的，但通常不连续。然后表明，在任何情况下，这两个函数在 $\lambda$ 拓扑中都是连续的。还介绍了CIFS解析族、常规解析族和平面解析族的概念。确定如果CIFS族是正则解析的，则Hausdorff维函数是实解析的；如果族是平面解析的，则豪斯多夫维数函数是连续且次调和的，但不一定是实解析的。然后将这些结果应用于有限抛物线 CIFS。进一步提供了反例，强调了分析族（但不是常规分析族）中豪斯多夫维度的实分析性细分。这些族经常表现出一种称为相变的现象。补充了防止这种转变发生的充分条件。
This paper deals with families of conformal iterated function systems (CIFS). The space of all CIFS, with common seed space X and alphabet I, is successively endowed with the topology of pointwise convergence and a new, weaker topology called $\lambda$-topology. It is proved that the pressure and the Hausdorff dimension of the limit set are continuous with respect to the topology of pointwise convergence when I is finite, and are lower semi-continuous, though generally not continuous, when I is infinite. It is then shown that these two functions are, in any case, continuous in the $\lambda$-topology. The concepts of analytic, regularly analytic and plane-analytic families of CIFS are also introduced. It is established that if a family of CIFS is regularly analytic, then the Hausdorff dimension function is real-analytic; if a family is plane-analytic, then the Hausdorff dimension function is continuous and subharmonic, though not necessarily real-analytic. These results are then applied to finite parabolic CIFS. Counter-examples highlighting breakdowns of real-analyticity in the Hausdorff dimension among analytic, but not regularly analytic, families are further provided. Such families often exhibit a phenomenon known as phase transition. Sufficient conditions preventing the occurrence of such transitions are supplemented.