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Boundary distance, lens
maps and entropy of geodesic flows of Finsler metrics
Dmitri Burago and Sergei Ivanov |
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Geometry & Topology 20 (2016) 469–490
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Abstract |
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We show that a small perturbation of the boundary distance function of a simple Finsler metric on the –disc is also the boundary distance function of some Finsler metric. (Simple metrics form an open class containing all flat metrics.) The lens map is a map that sends the exit vector to the entry vector as a geodesic crosses the disc. We show that a small perturbation of a lens map of a simple Finsler metric is in its turn the lens map of some Finsler metric. We use this result to construct a smooth perturbation of the metric on the standard –dimensional sphere to produce positive metric entropy of the geodesic flow. Furthermore, this flow exhibits local generation of metric entropy; that is, positive entropy is generated in arbitrarily small tubes around one trajectory. |
Keywords
Finsler metric, boundary distance, lens map, scattering
relation, Hamiltonian flow, perturbation, metric entropy
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Mathematical Subject Classification 2010
Primary: 53C60, 37A35, 37J40
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References |
Publication
Received: 7 August 2014
Revised: 8 October 2014
Accepted: 8 October 2014
Published: 29 February 2016
Proposed: Leonid Polterovich
Seconded: Jean-Pierre Otal, Gang Tian
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Authors |
| Dmitri Burago | |
| Department of Mathematics Pennsylvania State University University Park State College, PA 16802 USA |
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| Sergei Ivanov | |
| St. Petersburg Department of Steklov
Mathematical Institute Russian Academy of Sciences Fontanka 27 St. Petersburg 191023 Russia |
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