渐近可加集映射的可加实现
Additive realizations of asymptotically additive set maps
AI总结：
针对可数离散顺从群到巴拿赫空间的集映射，研究其渐近可加实现的条件，拓展了既有序列与单映射情形的结果，建立相对版本并推广热力学形式核心内容。
摘要：
给定一个可数离散顺从群，我们研究从该群到巴拿赫空间（或更一般地，完备半赋范空间）的集映射可被实现为某向量在群表示下的遍历和、且该实现与原映射渐近不可区分的条件。我们证明，对一致有界的群表示，该性质由有界渐近可加集映射类刻画，拓展了此前针对巴拿赫空间中的序列以及单个非扩张线性映射情形的研究成果。此外，我们建立了该刻画的相对版本，明确了何时可在指定目标集内选取可加实现。作为应用，我们的结果推广了热力学形式的核心内容，在可加与渐近可加框架之间搭建了桥梁。
英文摘要：
Given a countable discrete amenable group, we study conditions under which a set map into a Banach space (or more generally, a complete semi-normed space) can be realized as the ergodic sum of a vector under a group representation, such that the realization is asymptotically indistinguishable from the original map. We show that for uniformly bounded group representations, this property is characterized by the class of bounded asymptotically additive set maps, extending previous work for sequences in Banach spaces and on the case of a single non-expansive linear map. Additionally, we develop a relative version of this characterization, identifying when the additive realization can be chosen within a prescribed target set. As an application, our results generalize central aspects of thermodynamic formalism, bridging the additive and asymptotically additive frameworks.
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