发表机构
Australian Institute of Machine Learning; Adelaide University（澳大利亚机器学习研究所; 阿德莱德大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过Kajiwara-Watatani C*-代数研究大重叠迭代函数系统，引入马尔可夫划分构造内在Hutchinson测度，并完整分类KMS态，扩展了已有结果。
AI 中文摘要
我们通过Kajiwara-Watatani C*-代数及其KMS态研究具有大重叠的迭代函数系统。我们的主要贡献有三个方面。首先，我们为迭代函数系统引入马尔可夫划分，并给出一个算法，该算法在存在有限马尔可夫划分时构造一个有限马尔可夫划分。其次，对于承认有限马尔可夫划分的压缩系统，我们构造了Hutchinson测度的内在类比，该测度独立于映射的任何标记或加权。它是满足由系统的集值动力学决定的自相似积分方程的唯一Borel概率测度，并且当系统满足开集条件时，它恢复具有均匀权重的Hutchinson测度。最后，我们给出了每个此类系统的Kajiwara-Watatani代数上规范作用的KMS态的完整分类。临界逆温度是内在轨道的指数增长率，它可以严格小于映射数的自然对数，并且在该温度下的唯一KMS态由我们的内在Hutchinson测度给出。这些系统包括许多具有大重叠的系统，这些系统未被Izumi-Kajiwara-Watatani的结果所覆盖。
英文摘要
We study iterated function systems with large overlaps via the Kajiwara-Watatani C*-algebra and its KMS states. Our main contributions are threefold. First, we introduce Markov partitions for iterated function systems and give an algorithm that constructs a finite Markov partition whenever one exists. Second, for contractive systems admitting a finite Markov partition, we construct an intrinsic analogue of the Hutchinson measure that is independent of any labelling or weighting of the maps. It is the unique Borel probability measure satisfying a self-similar integral equation dictated by the set-valued dynamics of the system, and it recovers the Hutchinson measure with uniform weights when the system satisfies the open set condition. Finally, we give a complete classification of KMS states for the gauge action on the Kajiwara-Watatani algebra of every such system. The critical inverse temperature is the exponential growth rate of the intrinsic orbits, which can be strictly smaller than the natural logarithm of the number of maps, and the unique KMS state at this temperature is given by our intrinsic Hutchinson measure. These systems include many with large overlaps not covered by the results of Izumi-Kajiwara-Watatani.
Comments38 pages
