#### Volume 20, issue 1 (2016)

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
Boundary distance, lens maps and entropy of geodesic flows of Finsler metrics

### Dmitri Burago and Sergei Ivanov

Geometry & Topology 20 (2016) 469–490
##### Abstract

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.

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