Geodesic
Boundary distance, lens maps and entropy of geodesic flows of Finsler metrics
Burago, Dmitri1 ; Ivanov, Sergei2
1 Department of Mathematics, Pennsylvania State University, University Park, State College, PA 16802, USA
2 St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Geometry & topology, Tome 20 (2016) no. 1, pp. 469-490.

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

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.

DOI : 10.2140/gt.2016.20.469
Classification : 53C60, 37A35, 37J40
Keywords: Finsler metric, boundary distance, lens map, scattering relation, Hamiltonian flow, perturbation, metric entropy

Burago, Dmitri 1 ; Ivanov, Sergei 2

1 Department of Mathematics, Pennsylvania State University, University Park, State College, PA 16802, USA
2 St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia 
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Burago, Dmitri; Ivanov, Sergei. Boundary distance, lens maps and entropy of geodesic flows of Finsler metrics. Geometry & topology, Tome 20 (2016) no. 1, pp. 469-490. doi : 10.2140/gt.2016.20.469. http://geodesic.mathdoc.fr/articles/10.2140/gt.2016.20.469/

[1] A S Besicovitch, On two problems of Loewner, J. London Math. Soc. 27 (1952) 141

[2] R Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc. 164 (1972) 323

[3] D Y Burago, A new approach to the calculation of the entropy of the geodesic flow and of related dynamical systems, Dokl. Akad. Nauk SSSR 299 (1988) 1041

[4] S Ivanov, Local monotonicity of Riemannian and Finsler volume with respect to boundary distances, Geom. Dedicata 164 (2013) 83

[5] G Knieper, H Weiss, A surface with positive curvature and positive topological entropy, J. Differential Geom. 39 (1994) 229

[6] S E Newhouse, Continuity properties of entropy, Ann. of Math. 129 (1989) 215

[7] G P Paternain, Real analytic convex surfaces with positive topological entropy and rigid body dynamics, Manuscripta Math. 78 (1993) 397

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