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Consider a surface $S$ and a vector field on the surface which has a closed orbit. The vector field on both sides of the closed orbit spirals towards it, which gives us that the linearized Poincare return map has eigenvalue $\leq 1$. What additional information do I need to infer that the eigenvalue is strictly less than 1? I know that in a neighborhood of the closed orbit, the Gaussian curvature is negative. How are such conclusions generally made? Thanks a lot!

Edit after Ali Taghavi's comment: The integral curves of the vector field near the closed orbit have their accelerations ${\bf parallel}$ to the surface (this is kind of opposite of a geodesic).

Consider a surface $S$ and a vector field on the surface which has a closed orbit. The vector field on both sides of the closed orbit spirals towards it, which gives us that the linearized Poincare return map has eigenvalue $\leq 1$. What additional information do I need to infer that the eigenvalue is strictly less than 1? I know that in a neighborhood of the closed orbit, the Gaussian curvature is negative. How are such conclusions generally made? Thanks a lot!

Consider a surface $S$ and a vector field on the surface which has a closed orbit. The vector field on both sides of the closed orbit spirals towards it, which gives us that the linearized Poincare return map has eigenvalue $\leq 1$. What additional information do I need to infer that the eigenvalue is strictly less than 1? I know that in a neighborhood of the closed orbit, the Gaussian curvature is negative. How are such conclusions generally made? Thanks a lot!

Edit after Ali Taghavi's comment: The integral curves of the vector field near the closed orbit have their accelerations ${\bf parallel}$ to the surface (this is kind of opposite of a geodesic).

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user65812
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Unit eigenvalue of the linearized Poincare return map

Consider a surface $S$ and a vector field on the surface which has a closed orbit. The vector field on both sides of the closed orbit spirals towards it, which gives us that the linearized Poincare return map has eigenvalue $\leq 1$. What additional information do I need to infer that the eigenvalue is strictly less than 1? I know that in a neighborhood of the closed orbit, the Gaussian curvature is negative. How are such conclusions generally made? Thanks a lot!