发表机构
Universidade do Estado do Amapá - UEAP（阿马帕州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明连通的局部有限正则图及其与紧度量空间的乘积的超空间在豪斯多夫度量下是可塑的，并给出相关判据与反例。
AI 中文摘要
一个度量空间被称为可塑的，如果每个双射的非扩张自映射都是等距映射。我们证明了每个连通的、局部有限的、正则图的非空紧子集超空间，在配备与路径度量相关联的豪斯多夫度量下，是可塑的。该证明将具有最小单位球的集合识别为相邻孪生类的非空子集，并表明每个非扩张双射都会诱导商图的一个自同构，且该自同构保持这些类的大小。当不存在相邻孪生时，单点集被保持，但一般情况下不一定被保持。我们还证明了对于每个紧连通度量空间$K$和每个连通的、局部有限的、正则图$G$，在乘积上赋予上确界度量时，$\mathcal{K}(K\times G)$是可塑的，并给出了此类乘积的更一般判据。进一步的结果包括：一个可塑度量空间其超空间不可塑的例子、每棵树的不可塑性，以及一个连通的、局部有限的、只有有限个最小度顶点的图的超空间的可塑性。
英文摘要
A metric space is plastic if every bijective nonexpansive self-map is an isometry. We prove that the hyperspace of nonempty compact subsets of every connected, locally finite, regular graph, equipped with the Hausdorff metric associated with the path metric, is plastic. The proof identifies the sets with smallest unit balls as the nonempty subsets of adjacent-twin classes and shows that every nonexpansive bijection induces an automorphism of the quotient graph preserving the sizes of these classes. Singletons are preserved when there are no adjacent twins, but need not be preserved in general. We also prove that $\mathcal{K}(K\times G)$ is plastic for every compact connected metric space $K$ and every connected, locally finite, regular graph $G$, with the supremum metric on the product, and give a more general criterion for such products. Further results include a plastic metric space whose hyperspace is not plastic, plasticity of every tree, and plasticity of the hyperspace of a connected, locally finite graph with only finitely many vertices of minimum degree.
