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Mathematics > Dynamical Systems

arXiv:1405.0098 (math)
[Submitted on 1 May 2014 (v1), last revised 8 May 2014 (this version, v2)]

Title:Effective bounds in E.Hopf rigidity for billiards and geodesic flows

Authors:Michael (Misha)Bialy
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Abstract:In this paper we show that in some cases the this http URL rigidity phenomenon admits quantitative interpretation. More precisely we estimate from above the measure of the set $\mathcal{M}$ swept by minimal orbits. These estimates are sharp, i.e. if $\mathcal{M}$ occupies the whole phase space we recover the this http URL rigidity. We give these estimates in two cases: the first is the case of convex billiards in the plane, sphere or hyperbolic plane. The second is the case of conformally flat Riemannian metrics on a torus. It seems to be a challenging question to understand such a quantitative bounds for Burago-Ivanov theorem.
Comments: 11 pages
Subjects: Dynamical Systems (math.DS); Differential Geometry (math.DG)
Cite as: arXiv:1405.0098 [math.DS]
  (or arXiv:1405.0098v2 [math.DS] for this version)
  https://doi.org/10.48550/arXiv.1405.0098

Submission history

From: Michael (Misha) Bialy [view email]
[v1] Thu, 1 May 2014 05:19:22 UTC (9 KB)
[v2] Thu, 8 May 2014 08:51:33 UTC (9 KB)
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