Barcode entropy of geodesic flows
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866909427498483712 |
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| author | Ginzburg, Viktor L. Gurel, Basak Z. Mazzucchelli, Marco |
| author_facet | Ginzburg, Viktor L. Gurel, Basak Z. Mazzucchelli, Marco |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_00943 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Barcode entropy of geodesic flows Ginzburg, Viktor L. Gurel, Basak Z. Mazzucchelli, Marco Symplectic Geometry Differential Geometry Dynamical Systems 37D40, 37B40, 58E10 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. |
| title | Barcode entropy of geodesic flows |
| topic | Symplectic Geometry Differential Geometry Dynamical Systems 37D40, 37B40, 58E10 |
| url | https://arxiv.org/abs/2212.00943 |