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A Smooth Autonomous Flow with Exponential Mixing on the Three-Torus

  • Nianci Jiang
  • Weili Zhang
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Abstract

We construct a smooth autonomous divergence-free vector field on 𝕋3 whose flow is exponentially mixing with respect to Lebesgue measure. This answers an open question about smooth autonomous exponential mixing in three dimensions. Starting from an exponentially mixing Anosov suspension flow, we perform surgery near four periodic orbits and show that the resulting manifold is diffeomorphic to 𝕋3. The resulting vector field has exactly sixteen hyperbolic equilibria. We also prove a general transfer theorem that preserves exponential mixing under certain singular changes of coordinates.

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Cite

BibTeXRIS
BibTeX
@misc{jiang2026smooth,
  title = {{A Smooth Autonomous Flow with Exponential Mixing on the Three-Torus}},
  author = {Nianci Jiang and Weili Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15717},
  eprint = {2610.15717},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15717},
}
RIS
TY  - UNPB
TI  - A Smooth Autonomous Flow with Exponential Mixing on the Three-Torus
AU  - Nianci Jiang
AU  - Weili Zhang
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15717
UR  - https://arxiv.org/abs/2610.15717
AB  - We construct a smooth autonomous divergence-free vector field on 𝕋^3 whose flow is exponentially mixing with respect to Lebesgue measure. This answers an open question about smooth autonomous exponential mixing in three dimensions. Starting from an exponentially mixing Anosov suspension flow, we perform surgery near four periodic orbits and show that the resulting manifold is diffeomorphic to 𝕋^3. The resulting vector field has exactly sixteen hyperbolic equilibria. We also prove a general transfer theorem that preserves exponential mixing under certain singular changes of coordinates.
ER  -