Work

Analytic Liouville Integrable Geodesic Flow Has Zero Entropy

Year: 2026

Type: preprint

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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.16640

Authors: Liu, Fei, Liu, Xiaokai, Zhang, Xiang

Institutions: Shandong University of Science and Technology, Southern University of Science and Technology, Shanghai Jiao Tong University

Language: English

Topic: Mathematical Dynamics and Fractals

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Natural Science Foundation of China, Southern University of Science and Technology

Awards: G252719983, G8521977386

OpenAlex ID: W7221807906

Cites: 1

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Related to: 10