Geodesic
Structural equivalence of mathematical billiards
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2009), pp. 66-69

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It is shown, that under the assumption on the smoothness and strong convexity of boundaries, the mathematical billiards are structurally equivalent if and only if their regions are geometrically similar and the equivalence mapping of the boundaries is a similarity mapping. 
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     title = {Structural equivalence of mathematical billiards},
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L. V. Butorina; E. I. Kugushev. Structural equivalence of mathematical billiards. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2009), pp. 66-69. http://geodesic.mathdoc.fr/item/VMUMM_2009_5_a15/