无限系统中的量子化系数
Quantization coefficients in infinite systems
摘要：
本文研究了极限集上的自相似概率测度μ的量化系数，这些极限集是由具有无穷多个压缩相似性的系统S和概率向量生成的.无限系统的量子化系数理论与有限系统的量子化系数理论有很大的不同。其中一个区别是缺乏有限极大反链，以及收缩比集的下确界为零;另一个区别在于S的非紧极限集J的特定几何。证明了对每个r \in（0，1），存在唯一的正数κ_r，使得对任意的κ< κ_r <κ '，μ的k维r阶下量子化系数为正，并给出了μ的κ'维r阶上量子化系数的估计.特别地，由此得出μ的r阶量子化维数存在，并且它等于κ_r。上述结果允许估计L_r-Kantorovich-Wasserstein度量中测度μ的渐近误差，其中离散测度支持在多个点上.
英文摘要：
We investigate quantization coefficients for self-similar probability measures μon limit sets which are generated by systems S of infinitely many contractive similarities and by probabilistic vectors. The theory of quantization coefficients for infinite systems has significant differences from the finite case. One of these differences is the lack of finite maximal antichains, and the fact that the set of contraction ratios has zero infimum; another difference resides in the specific geometry of the non-compact limit set J of S. We prove that, for each r \in (0,1), there exists a unique positive number κ_r, so that for arbitrary κ< κ_r < κ', the κ-dimensional lower quantization coefficient of order r of μis positive, and we also give estimates for the κ'-dimensional upper quantization coefficient of order r of μ. In particular, it follows that the quantization dimension of order r of μexists, and it is equal to κ_r. The above results allow then to estimate the asymptotic errors of approximating the measure μin the L_r-Kantorovich-Wasserstein metric, with discrete measures supported on finitely many points.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
