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PreprintCC BY 4.0arXiv

Ergodicity of geodesic flows on Aravinda-Farrell twisted doubles

  • Alberto Verjovsky
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Abstract

The announced rank-one ergodicity theorem of Rodriguez Hertz, Rodriguez Hertz, and Wu includes the nonpositively curved metrics considered here on Aravinda--Farrell twisted doubles. For these construction metrics, a direct geometric argument identifies the singular set. Their singular geodesics are precisely those tangent to the torus sections of finitely many flat product collars, so their initial vectors have Liouville measure zero. A compact simply connected modification region cannot contain a complete geodesic; this also treats the exotic connected-sum examples. Ergodicity then follows from the classical theorem on the regular set.

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Cite

BibTeXRIS
BibTeX
@misc{verjovsky2026ergodicity,
  title = {{Ergodicity of geodesic flows on Aravinda-Farrell twisted doubles}},
  author = {Alberto Verjovsky},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.02106},
  eprint = {2611.02106},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.02106},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Ergodicity of geodesic flows on Aravinda-Farrell twisted doubles
AU  - Alberto Verjovsky
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.02106
UR  - https://arxiv.org/abs/2611.02106
AB  - The announced rank-one ergodicity theorem of Rodriguez Hertz, Rodriguez Hertz, and Wu includes the nonpositively curved metrics considered here on Aravinda--Farrell twisted doubles. For these construction metrics, a direct geometric argument identifies the singular set. Their singular geodesics are precisely those tangent to the torus sections of finitely many flat product collars, so their initial vectors have Liouville measure zero. A compact simply connected modification region cannot contain a complete geodesic; this also treats the exotic connected-sum examples. Ergodicity then follows from the classical theorem on the regular set.
ER  -