Asian Journal of Mathematics

Volume 25 (2021)

Number 5

Finsler perturbation with nondense geodesics with irrational directions

Pages: 715 – 726

DOI: https://dx.doi.org/10.4310/AJM.2021.v25.n5.a5

Authors

Dmitri Burago (Department of Mathematics, Pennsylvania State University, University Park, Pa., U.S.A.)

Dong Chen (Department of Mathematics, Pennsylvania State University, University Park, Pa., U.S.A.)

Abstract

We show that given any Liouville direction and flat Finsler torus, one can make a $C^\infty$‑small perturbation on an arbitrarily small disc to get a nondense geodesic in the given direction.

Keywords

Finsler geometry, perturbation, nondense geodesics, dual lens maps, Aubry–Mather theory

2010 Mathematics Subject Classification

53C23, 53C60

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The first-named author was partially supported by NSF grant DMS-1205597.

The second-named author was partially supported by NSF grant DMS-1205597 and D. Burago’s PSU Research Fund.

Received 18 February 2019

Accepted 17 June 2021

Published 6 July 2022