On ergodic properties of certain billiards
On ergodic properties of certain billiards
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关于某些台球的遍历特性
DOI：
10.1007/bf01075700
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发表时间：
1974
影响因子：
0.4
通讯作者：
L. Bunimovich
中科院分区：
文献类型：
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作者：
L. Bunimovich
* * 它：众所周知（参见[1]（2）：分散台球（即：位于二维环面T2或欧氏平面R2上的区域Q（其边界由从Q向内凸的分量组成）中的台球是Kolmogorov意义下的K系统。文[5]对一类边界同时含有色散分量和聚焦分量的区域QcR2也得到了类似的结果。另一方面，具有足够光滑边界的凸区域中的台球不是遍历的（参见，例如，[6]）.本文的目的是证明存在凸区域QcR2使得Q中的台球是K系统.对于第六章，Arnol'd [7]给出了负曲率曲面上离散面元与测地线流之间的类比的图解说明。在这个意义上，我们所考虑的台球类似于曲率不为负的曲面上的测地线流。
::: It: is known (cf.[1] and [2]) that: dispersing billiards (ie,: billiards in regions Q, located on a two-dimensional torus T 2 or on the Euclidean plane R 2, whose boundary consists of components convex inward from Q) are K systems in the sense Of AN Kolmogorov [3, 4]. In [5], an analogous result was obtained for one class of regions Q c R 2 whose boundaries contain both dispersing and focusing components. On the other hand, billiards in a convex region with sufficiently smooth boundaries are not ergodic (cf., for example,[6]).The aim of the present note is to show that there exist convex regions Q c R 2 such that a billiard in Q is a K system. To VI Arnol'd [7] is due the graphic explanation of the analogy between dispersing bin liards and geodesic flows on surfaces of negative curvature. In this sense, the billiards considered by us are analogous to geodesic flows on surfaces whose curvature is not negative.