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Every ergodic measure is uniquely maximizing
Insecure configurations in lattice translation surfaces, with applications to polygonal billiards
1.  University of California and IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brasil 22460320, Brazil 
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.
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