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

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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BibTeX RIS

BibTeX

@misc{liu2026measure,
  title = {{Measure of Maximal Entropy for Geodesic Flows on Noncompact Manifolds of Nonpositive Curvature}},
  author = {Fei Liu and Xiaokai Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37439},
  eprint = {2610.37439},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37439},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Measure of Maximal Entropy for Geodesic Flows on Noncompact Manifolds of Nonpositive Curvature
AU  - Fei Liu
AU  - Xiaokai Liu
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37439
UR  - https://arxiv.org/abs/2610.37439
AB  - 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.
ER  -

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