Characterization of geodesic flows on \(T^{2}\) with and without positive topological entropy
The authors consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus for an arbitrary Riemannian metric. They formulate necessary and sufficient conditions that the topological entropy be zero, and show that in the case of zero entropy the flow has properties like those of integrable systems. Specifically, they prove the existence of a continuous constant of motion that measures the direction of geodesics lifted from \(T^2\) to \(\mathbb{R}^2\).
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