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Boundary distance, lens maps and entropy of geodesic flows of Finsler metrics

Dmitri Burago and Sergei Ivanov

Geometry & Topology 20 (2016) 469–490

We show that a small perturbation of the boundary distance function of a simple Finsler metric on the n–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 4–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.

Finsler metric, boundary distance, lens map, scattering relation, Hamiltonian flow, perturbation, metric entropy
Mathematical Subject Classification 2010
Primary: 53C60, 37A35, 37J40
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
Dmitri Burago
Department of Mathematics
Pennsylvania State University
University Park
State College, PA 16802
Sergei Ivanov
St. Petersburg Department of Steklov Mathematical Institute
Russian Academy of Sciences
Fontanka 27
St. Petersburg 191023