Abstract
We prove that for a broad class of exact symplectic manifolds including \({\mathbb{R}^{2m}}\), the Hamiltonian flow on a regular compact energy level of an autonomous Hamiltonian cannot be uniquely ergodic. This is a consequence of the Weinstein conjecture and the observation that a Hamiltonian structure with non-vanishing self-linking number must have contact type. We apply these results to show that certain types of exact twisted geodesic flows cannot be uniquely ergodic.
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The work is partially supported by NSF Grant DMS-1308501.
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Ginzburg, V.L., Niche, C.J. A remark on unique ergodicity and the contact type condition. Arch. Math. 105, 585–592 (2015). https://doi.org/10.1007/s00013-015-0832-8
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DOI: https://doi.org/10.1007/s00013-015-0832-8