June  2006, 16(2): 367-382. doi: 10.3934/dcds.2006.16.367

Insecure configurations in lattice translation surfaces, with applications to polygonal billiards


University of California and IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brasil 22460-320, Brazil

Received  March 2005 Revised  December 2005 Published  July 2006

A configuration (i.e., a pair of points) in a Riemannian space $X$ is secure if all connecting geodesics can be blocked by a finite subset of $X$. A space is secure if all of its configurations are secure. Secure spaces seem to be rare.
    If $X$ is an insecure space, it is natural to ask how big the set of insecure configurations is. We investigate this problem for flat surfaces, in particular for translation surfaces and polygons, from the viewpoint of measure theory.
    Here is a sample of our results. Let $X$ be a lattice translation surface or a lattice polygon. Then the following dichotomy holds: i) The surface (polygon) $X$ is arithmetic. Then all configurations in $X$ are secure; ii) The surface (polygon) $X$ is nonarithmetic. Then almost all configurations in $X$ are insecure.
Citation: Eugene Gutkin. Insecure configurations in lattice translation surfaces, with applications to polygonal billiards. Discrete & Continuous Dynamical Systems, 2006, 16 (2) : 367-382. doi: 10.3934/dcds.2006.16.367

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