论文标题
在带有三角轨道开放子集的投射台球上
On projective billiards with open subsets of triangular orbits
论文作者
论文摘要
IVRII的猜想指出，在欧几里得空间中的每个台球中，周期性轨道集的测量都为零。这意味着每$ k \ geq2 $都没有K反射台球，即具有开放式k骨轨道的台球。这种猜想在欧几里得空间中开放，只有很少的部分结果。众所周知，在二维球中，存在3个反射台球（Yu.M.Baryshnikov）。所有三个反射的球形台球都被V.Blumen，K.Kim，J.Nance，V.Zharnitsky归类为：它们的边界位于三个正交大圈子中。在本文中，我们研究了S.Tabachnikov引入的投射台球的IVRII猜想的类似物。在两个维度上，存在一个3反射的投影台球，即所谓的右手台球，这是球形3式柱面的预测。我们表明，唯一具有分段平滑边界的3型平面投影台球是上述右台球。在较高的维度中，我们证明了具有分段平滑边界的3反射投影台球的不存在，并且具有分段平滑边界的投射台球的不存在，具有相位空间中非零测度的三角形轨道的子集。
Ivrii's Conjecture states that in every billiard in Euclidean space the set of periodic orbits has measure zero. It implies that for every $k\geq2$ there are no k-reflective billiards, i.e., billiards having an open set of k-periodic orbits. This conjecture is open in Euclidean spaces, with just few partial results. It is known that in the two-dimensional sphere there exist 3-reflective billiards (Yu.M.Baryshnikov). All the 3-reflective spherical billiards were classified in a paper by V.Blumen, K.Kim, J.Nance, V.Zharnitsky: the boundary of each of them lies in three orthogonal big circles. In the present paper we study the analogue of Ivrii's Conjecture for projective billiards introduced by S.Tabachnikov. In two dimensions there exists a 3-reflective projective billiard, the so-called right-spherical billiard, which is the projection of a spherical 3-reflective billiard. We show that the only 3-reflective planar projective billiard with piecewise smooth boundary is the above-mentioned right-spherical billiard. In higher dimensions, we prove the non-existence of 3-reflective projective billiards with piecewise smooth boundary, and also the non-existence of projective billiards with piecewise smooth boundary having a subset of triangular orbits of non-zero measure in the phase space.
