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PreprintCC BY 4.0arXiv

Typical expansive measures for infinite-dimensional maps

  • Gabriel Lacerda
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Abstract

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 Gδ subset, revealing a rich measure-theoretic structure despite the infinite dimensionality of the phase space.

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Cite

BibTeXRIS
BibTeX
@misc{lacerda2026typical,
  title = {{Typical expansive measures for infinite-dimensional maps}},
  author = {Gabriel Lacerda},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.09482},
  eprint = {2610.09482},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.09482},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Typical expansive measures for infinite-dimensional maps
AU  - Gabriel Lacerda
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.09482
UR  - https://arxiv.org/abs/2610.09482
AB  - 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.
ER  -