发表机构
Universidade Federal do Rio de Janeiro（里约热内卢联邦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文否定回答了Morales和Sirvent关于膨胀性维度约束是否扩展到测度论概念的问题，证明Hilbert立方体上的移位映射关于乘积Lebesgue测度是膨胀的，并建立了一般充分条件，揭示了无限维相空间中丰富的测度论结构。
AI 中文摘要
Mañé的一个经典定理指出，允许膨胀同胚的紧致度量空间必须是有限维的。Morales和Sirvent提出了一个自然问题：这个维度约束是否扩展到膨胀性的测度论概念。在本文中，我们对这个问题给出了否定回答。我们证明了Hilbert立方体上的移位映射关于乘积Lebesgue测度是膨胀的。此外，我们建立了一个一般的充分条件：任何允许全支撑膨胀测度的同胚都是稠密测度膨胀的。作为推论，我们证明了移位映射的膨胀且拓扑稳定的测度集合包含一个稠密的$G_\delta$子集，揭示了尽管相空间是无限维的，但测度论结构是丰富的。
英文摘要
A classical theorem by Mañé states that compact metric spaces admitting expansive homeomorphisms must be finite-dimensional. It is a natural question, raised by Morales and Sirvent, whether this dimensional constraint extends to the measure-theoretic notion of expansiveness. In this paper, we answer this question in the negative. We prove that the shift map on the Hilbert cube is expansive with respect to the product Lebesgue measure. Furthermore, we establish a general sufficient condition: any homeomorphism admitting a full-support expansive measure is densely measure-expansive. As a consequence, we show that the set of expansive and topologically stable measures for the shift map contains a dense $G_δ$ subset, revealing a rich measure-theoretic structure despite the infinite dimensionality of the phase space.
