论文标题
无限自相似措施的几何均值误差的量化维度和稳定性
Quantization dimension and stability for infinite self-similar measures with respect to geometric mean error
论文作者
论文摘要
令$μ$为与迭代功能系统相关的BOREL概率度量，该功能系统由数量无限的合同相似性和无限概率向量组成。在本文中，我们研究了度量$μ$相对于几何均值误差的量化维度。无限系统的量化与Graf和Luschgy研究的众所周知的有限案例不同。也就是说，在有限设置中使用的许多工具，例如，有限的最大抗大众存在在无限情况下失败。我们证明，度量$μ$的量化维度等于其Hausdorff尺寸，该维度扩展了有限情况的Graf和Luschgy的众所周知的结果到无限设置。在最后一节中，我们讨论了无限系统量化维度的稳定性。
Let $μ$ be a Borel probability measure associated with an iterated function system consisting of a countably infinite number of contracting similarities and an infinite probability vector. In this paper, we study the quantization dimension of the measure $μ$ with respect to the geometric mean error. The quantization for infinite systems is different from the well-known finite case investigated by Graf and Luschgy. That is, many tools which are used in the finite setting, for example, existence of finite maximal antichains, fail in the infinite case. We prove that the quantization dimension of the measure $μ$ is equal to its Hausdorff dimension which extends a well-known result of Graf and Luschgy for the finite case to an infinite setting. In the last section, we discuss the stability of quantization dimension for infinite systems.
