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arXiv · 2610.37439

Measure of Maximal Entropy for Geodesic Flows on Noncompact Manifolds of Nonpositive Curvature

Fei Liu·Xiaokai Liu
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Abstract

We establish the variational characterization of the measure of maximal entropy on noncompact manifolds of nonpositive sectional curvature. Assuming that the deck transformation group contains an axial isometry with a rank-one axis and has a finite critical exponent, we prove that the supremum of the entropies of invariant probabilities equals the critical exponent and is attained if and only if the Bowen-Margulis-Sullivan (BMS) measure is finite. In that case, the measure of maximal entropy is unique and equals the normalized BMS measure. The proof combines conditional-entropy rigidity on the regular set with an HTS-based entropy exclusion on the singular set, overcoming both non-compactness and the loss of uniform hyperbolicity due to zero curvature. We also obtain mixing under a non-arithmetic length-spectrum assumption. For nonpositively curved surfaces of genus at least two, the existence of a closed Jacobi rank-one geodesic and the non-arithmeticity of the axial length spectrum are automatic.

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Fei Liu, Xiaokai Liu. 2026-10-27. Measure of Maximal Entropy for Geodesic Flows on Noncompact Manifolds of Nonpositive Curvature. https://arxiv.org/abs/2610.37439

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