带小孔台球的逃脱率和物理相关措施
Escape Rates and Physically Relevant Measures for Billiards with Small Holes
摘要：
本文研究了二维周期洛仑兹气体在表中存在小孔时的台球映射。我们允许孔的形式开集远离散射体以及段的边界上的散射体。对于一大类光滑初始分布，我们建立了一个共同的逃逸率和归一化极限分布的存在性。这种极限分布是条件不变的，是一个封闭系统的SRB措施的自然模拟。最后，我们证明了当洞的大小趋于零时，极限分布收敛到台球映射的光滑不变测度。
英文摘要：
We study the billiard map corresponding to a periodic Lorentz gas in 2-dimensions in the presence of small holes in the table. We allow holes in the form of open sets away from the scatterers as well as segments on the boundaries of the scatterers. For a large class of smooth initial distributions, we establish the existence of a common escape rate and normalized limiting distribution. This limiting distribution is conditionally invariant and is the natural analogue of the SRB measure of a closed system. Finally, we prove that as the size of the hole tends to zero, the limiting distribution converges to the smooth invariant measure of the billiard map.
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