Existence of a positive hyperbolic orbit in three-dimensional Reeb flows
Abstract
Non-degenerate periodic orbits in three-dimensional Reeb flows are classified into three types: positive hyperbolic, negative hyperbolic and elliptic. In the present paper, we consider a closed connected contact three-manifold with a non-degenerate contact form. We show that its Reeb flow has a simple positive hyperbolic orbit if it has at least three simple periodic orbits. We mainly study the case in which no elliptic orbit exists. We prove that there is no non-degenerate contact form on a closed connected three-manifold with b1=0 such that all simple periodic orbits are negative hyperbolic. As a corollary, by combining the author's previous result in the presence of an elliptic orbit and the known result for b1>0, we obtain the main result. The proof uses the Weyl law for ECH spectral invariants. We also use compactness for genus zero J-holomorphic curves counted by the U-map. Under the contrary assumption, the number of simple orbits in each action interval [L,2L] is uniformly bounded with respect to L. We use this property to study ECH generators and genus zero U-curves.
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BibTeX
@misc{shibata2026existence,
title = {{Existence of a positive hyperbolic orbit in three-dimensional Reeb flows}},
author = {Taisuke Shibata},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16176},
eprint = {2609.16176},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16176},
}
RIS
TY - UNPB
TI - Existence of a positive hyperbolic orbit in three-dimensional Reeb flows
AU - Taisuke Shibata
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.16176
UR - https://arxiv.org/abs/2609.16176
AB - Non-degenerate periodic orbits in three-dimensional Reeb flows are classified into three types: positive hyperbolic, negative hyperbolic and elliptic. In the present paper, we consider a closed connected contact three-manifold with a non-degenerate contact form. We show that its Reeb flow has a simple positive hyperbolic orbit if it has at least three simple periodic orbits. We mainly study the case in which no elliptic orbit exists. We prove that there is no non-degenerate contact form on a closed connected three-manifold with b_1=0 such that all simple periodic orbits are negative hyperbolic. As a corollary, by combining the author's previous result in the presence of an elliptic orbit and the known result for b_1>0, we obtain the main result. The proof uses the Weyl law for ECH spectral invariants. We also use compactness for genus zero J-holomorphic curves counted by the U-map. Under the contrary assumption, the number of simple orbits in each action interval [L,2L] is uniformly bounded with respect to L. We use this property to study ECH generators and genus zero U-curves.
ER -