Abstract
We study the size of the set of ergodic directions for the directional billiard flows on the infinite band \({\mathbb{R}\times [0,h]}\) with periodically placed linear barriers of length 0 < λ < h. We prove that the set of ergodic directions is always uncountable. Moreover, if λ/h ∈ (0, 1) is rational, the Hausdorff dimension of the set of ergodic directions is greater than 1/2. In both cases (rational and irrational) we construct explicitly some sets of ergodic directions.
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Communicated by M. Lyubich
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Frączek, K., Ulcigrai, C. Ergodic Directions for Billiards in a Strip with Periodically Located Obstacles. Commun. Math. Phys. 327, 643–663 (2014). https://doi.org/10.1007/s00220-014-2017-x
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DOI: https://doi.org/10.1007/s00220-014-2017-x