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Self-intersecting Geodesics and Entropy of the Geodesic Flow

Summary: Self-intersecting Geodesics
and Entropy of the Geodesic Flow
Sigurd Angenent
This note was inspired by a paper of Denvir and MacKay [1] who showed
(among other things) that the geodesic flow on a torus T2
with Riemannian metric
g has positive topological entropy as soon as it admits a contractible closed geodesic.
Our main observation concerns closed geodesics on surfaces M with a smooth
Finsler metric, i.e. a function F : T M [0, ) which is a norm on each tangent
space TpM, p M, which is smooth outside of the zero section in T M, and which
is strictly convex in the sense that Hess(F2
) is positive definite on TpM \ {0}.
One calls a Finsler metric F symmetric if F(p, -v) = F(p, v) for all v TpM.
We denote the universal cover of a surface M by ^M. Any Finsler metric on M lifts
to a Finsler metric on ^M which we again denote by F
Lemma. Let M be a compact surface with (M) 0, and let F : T M R
be a smooth symmetric Finsler metric on M. If the lift ^ : R ^M of some closed
geodesic : R M has a self intersection, then ( ^M, F) admits a simple closed
geodesic whose projection to M is thus a contractible closed geodesic for (M, F).
Since the Denvir-MacKay result generalizes to the case of Finsler metrics (see


Source: Angenent, Sigurd - Department of Mathematics, University of Wisconsin at Madison


Collections: Mathematics