Abstract
We prove that the natural invariant surface associated with the billiard game on an irrational polygonal table is homeomorphic to the Loch Ness monster, that is, the only orientable infinite genus topological real surface with exactly one end.
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Acknowledgments
The author gratefully acknowledges support from Sonderforschungsbereich/Transregio 45 and ANR Symplexe and thanks the Max-Planck Institut für Mathematik in Bonn for the excellent working conditions provided during the preparation of this paper. The author is indebted to M. Möller, B. Weiss for useful comments and discussions, and would like to express his gratitude to the referee for many valuable remarks.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Valdez, F. Infinite genus surfaces and irrational polygonal billiards. Geom Dedicata 143, 143–154 (2009). https://doi.org/10.1007/s10711-009-9378-x
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DOI: https://doi.org/10.1007/s10711-009-9378-x