Note on the periodic points of the billiard (Q1189296)

Let \(\Omega\) be a bounded domain in the Euclidean plane with smooth boundary \(\partial \Omega\). Denote by \(\text{Fix}_ 3\) the set of all 3- periodic points of the billiard ball map related to \(\Omega\). The author proves that if \(\partial \Omega\) is of the class \(C^ 3\), then \(\text{Fix}_ 3\) has empty interior and the Lebesgue measure zero in its (two dimensional) phase space. This generalizes formally a result of \textit{M. Rychlik} [ibid. 30, No. 1, 191-205 (1989; Zbl 0678.58035)] (referred as [2] in the paper) who assumed that \(\Omega\) is also convex. In the proof of the author's theorem the notation of [2] is used; the main difference between the author's proof and that of [2] is in the method applied in the last parts of them: the reasoning proposed by the author is much more elementary.