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Hyperbolic periodic orbits and optimal 1/t convergence of the Lax--Oleinik semigroup

  • Qinbo Chen
  • Zhi-Xiang Zhu
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Abstract

Let M be a closed manifold and H:T*M→ℝ a Ck (k≥2) Tonelli Hamiltonian whose Aubry set consists of finitely many hyperbolic orbits, each of which is either a periodic orbit or a fixed point. We construct a Ck critical subsolution that is quadratically strict outside the projected Aubry set. We then use this subsolution to establish an O(1/t) rate of convergence for the associated Lax--Oleinik semigroup and the minimal action functions. This rate is optimal whenever at least one of these hyperbolic orbits is periodic.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026hyperbolic,
  title = {{Hyperbolic periodic orbits and optimal $1/t$ convergence of the Lax--Oleinik semigroup}},
  author = {Qinbo Chen and Zhi-Xiang Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.03505},
  eprint = {2611.03505},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.03505},
}
RIS
TY  - UNPB
TI  - Hyperbolic periodic orbits and optimal 1/t convergence of the Lax--Oleinik semigroup
AU  - Qinbo Chen
AU  - Zhi-Xiang Zhu
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.03505
UR  - https://arxiv.org/abs/2611.03505
AB  - Let M be a closed manifold and H:T^*M→ℝ a C^k (k≥2) Tonelli Hamiltonian whose Aubry set consists of finitely many hyperbolic orbits, each of which is either a periodic orbit or a fixed point. We construct a C^k critical subsolution that is quadratically strict outside the projected Aubry set. We then use this subsolution to establish an O(1/t) rate of convergence for the associated Lax--Oleinik semigroup and the minimal action functions. This rate is optimal whenever at least one of these hyperbolic orbits is periodic.
ER  -