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On the Finiteness of Anosov flows on $3$-manifolds via Contact Geometry

  • Jonathan Bowden
  • Vincent Colin
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Abstract

In dimension 3, following Eliashberg-Thurston and Mitsumatsu, one can associate a transverse pair of oppositely oriented contact structures to any Anosov flow. We show that when the ambient manifold is hyperbolic, the isotopy classes of these contact structures completely determine the flow up to isotopy orbit equivalence. This then implies that on any closed hyperbolic 3-manifold, the number of Anosov flows modulo isotopy orbit equivalence is finite. This approach also yields an explicit bound on the number of orbit equivalence classes of Anosov flows in terms of the number of universally tight contact structures. In addition, we show finiteness of pseudo-Anosov flows for which the complement of the singular orbits is atoroidal and, in the non- hyperbolic case, we obtain a control on the dynamics of Anosov flows on the hyperbolic pieces of the JSJ-decomposition.

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Cite

BibTeXRIS
BibTeX
@misc{bowden2026finiteness,
  title = {{On the Finiteness of Anosov flows on $3$-manifolds via Contact Geometry}},
  author = {Jonathan Bowden and Vincent Colin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.03700},
  eprint = {2609.03700},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03700},
}
RIS
TY  - UNPB
TI  - On the Finiteness of Anosov flows on $3$-manifolds via Contact Geometry
AU  - Jonathan Bowden
AU  - Vincent Colin
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.03700
UR  - https://arxiv.org/abs/2609.03700
AB  - In dimension 3, following Eliashberg-Thurston and Mitsumatsu, one can associate a transverse pair of oppositely oriented contact structures to any Anosov flow. We show that when the ambient manifold is hyperbolic, the isotopy classes of these contact structures completely determine the flow up to isotopy orbit equivalence. This then implies that on any closed hyperbolic 3-manifold, the number of Anosov flows modulo isotopy orbit equivalence is finite. This approach also yields an explicit bound on the number of orbit equivalence classes of Anosov flows in terms of the number of universally tight contact structures. In addition, we show finiteness of pseudo-Anosov flows for which the complement of the singular orbits is atoroidal and, in the non- hyperbolic case, we obtain a control on the dynamics of Anosov flows on the hyperbolic pieces of the JSJ-decomposition.
ER  -