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Mathematics > Symplectic Geometry

arXiv:2212.00943 (math)
[Submitted on 2 Dec 2022 (v1), last revised 11 Dec 2024 (this version, v2)]

Title:Barcode entropy of geodesic flows

Authors:Viktor L. Ginzburg, Basak Z. Gurel, Marco Mazzucchelli
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Abstract:We introduce and study the barcode entropy for geodesic flows of closed Riemannian manifolds, which measures the exponential growth rate of the number of not-too-short bars in the Morse-theoretic barcode of the energy functional. We prove that the barcode entropy bounds from below the topological entropy of the geodesic flow and, conversely, bounds from above the topological entropy of any hyperbolic compact invariant set. As a consequence, for Riemannian metrics on surfaces, the barcode entropy is equal to the topological entropy. A key to the proofs and of independent interest is a crossing energy theorem for gradient flow lines of the energy functional.
Comments: 41 pages; final version: minor corrections. To appear in Journal of the European Mathematical Society
Subjects: Symplectic Geometry (math.SG); Differential Geometry (math.DG); Dynamical Systems (math.DS)
MSC classes: 37D40, 37B40, 58E10
Cite as: arXiv:2212.00943 [math.SG]
  (or arXiv:2212.00943v2 [math.SG] for this version)
  https://doi.org/10.48550/arXiv.2212.00943
Related DOI: https://doi.org/10.4171/JEMS/1572

Submission history

From: Marco Mazzucchelli [view email]
[v1] Fri, 2 Dec 2022 03:05:44 UTC (43 KB)
[v2] Wed, 11 Dec 2024 22:10:30 UTC (44 KB)
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