Analytic Liouville Integrable Geodesic Flow Has Zero Entropy

Abstract

This paper studies the topological entropy of real-analytic Liouville integrable Hamiltonian systems. We prove that the Hamiltonian flow on a compact connected regular energy hypersurface has zero topological entropy whenever a complete family of commuting real-analytic first integrals is generically independent in a connected neighborhood of that hypersurface. The proof uses finite normal expansions along minimal analytic carriers and an ergodic version of Ziglin's leading-term argument to obtain independent invariant normal polynomials, whose Poisson geometry makes all tangent Lyapunov exponents of the carriers vanish. In particular, we prove that a real-analytic Liouville integrable geodesic flow on a closed real-analytic Riemannian manifold has zero topological entropy, giving a positive answer to the conjecture proposed by Paternain in 1992.

Integrable and non-integrable LR geodesic flows on 5-dimensional 2-step nilpotent Lie groups
Milan Pavlović, Tijana Šukilović
arXiv29 Oct 2026PreprintCC BY-NC-ND 4.0Full textDynamical systems Geometry & topologyarXiv
Transmutation operators for the Sturm--Liouville equation in impedance form with an integrable potential
Briceyda B. Delgado, F. Ayça Çetinkaya
arXiv23 Sep 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
A Twisted Affine Interpolation of the ℤ₂²-graded Integrable Super-Liouville and super-Sinh-Gordon Equations
Naruhiko Aizawa, Ren Ito, Luiza Miranda +1
arXiv24 Oct 2026PreprintarXiv licenceFull textMathematical physics High-energy & particle physicsarXiv
Arithmetic Rigidity of Analytic Functions and Mahler's Problem on Liouville Numbers
Diego Marques
arXiv7 Oct 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
Measure of Maximal Entropy for Geodesic Flows on Noncompact Manifolds of Nonpositive Curvature
Fei Liu, Xiaokai Liu
arXiv27 Oct 2026PreprintCC BY 4.0Full textDynamical systems Geometry & topologyarXiv

Cite

BibTeX RIS

BibTeX

@misc{liu2026analytic,
  title = {{Analytic Liouville Integrable Geodesic Flow Has Zero Entropy}},
  author = {Fei Liu and Xiaokai Liu and Xiang Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16640},
  eprint = {2610.16640},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16640},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Analytic Liouville Integrable Geodesic Flow Has Zero Entropy
AU  - Fei Liu
AU  - Xiaokai Liu
AU  - Xiang Zhang
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16640
UR  - https://arxiv.org/abs/2610.16640
AB  - This paper studies the topological entropy of real-analytic Liouville integrable Hamiltonian systems. We prove that the Hamiltonian flow on a compact connected regular energy hypersurface has zero topological entropy whenever a complete family of commuting real-analytic first integrals is generically independent in a connected neighborhood of that hypersurface. The proof uses finite normal expansions along minimal analytic carriers and an ergodic version of Ziglin's leading-term argument to obtain independent invariant normal polynomials, whose Poisson geometry makes all tangent Lyapunov exponents of the carriers vanish. In particular, we prove that a real-analytic Liouville integrable geodesic flow on a closed real-analytic Riemannian manifold has zero topological entropy, giving a positive answer to the conjecture proposed by Paternain in 1992.
ER  -

Abstract

Related

Cite