一类具有重叠的自相似测度的量化误差的渐近阶
Asymptotic order of the quantization error for a class of self-similar measures with overlaps
AI总结：
本文研究满足有限型条件的等压缩相似映射生成的自相似测度的渐近量化误差，在可验证假设下证明其上、下量化系数均有界，推广了开集条件下的经典结果并适用于 Erdös 测度等重叠情形。
摘要：
设 $\{f_i\}_{i=1}^N$ 是 $\mathbb{R}^1$ 上满足有限型条件的一族等压缩相似映射。我们研究与 $\{f_i\}_{i=1}^N$ 及一个正概率向量相关联的自相似测度 $μ$ 的渐近量化误差。在一个可验证的假设下，我们证明了 $μ$ 的上、下量化系数都既远离零也远离无穷。这可以看作是对 Graf 和 Luschgy 关于具有开集条件的自相似测度结果的推广。我们的结果适用于一大类具有重叠的自相似测度，包括 Erdös 测度、经典 Cantor 测度的 $3$ 重卷积以及某些 $λ$-Cantor 集上的自相似测度。
英文摘要：
Let $\{f_i\}_{i=1}^N$ be a set of equi-contractive similitudes on $\mathbb{R}^1$ satisfying the finite-type condition. We study the asymptotic quantization error for self-similar measures $μ$ associated with $\{f_i\}_{i=1}^N$ and a positive probability vector. With a verifiable assumption, we prove that the upper and lower quantization coefficient for $μ$ are both bounded away from zero and infinity. This can be regarded as an extension of Graf and Luschgy's result on self-similar measures with the open set condition. Our result is applicable to a significant class of self-similar measures with overlaps, including Erdös measure, the $3$-fold convolution of the classical Cantor measure and the self-similar measures on some $λ$-Cantor sets.
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