论文标题
具有多个停止点的非理性流动的历史行为与身体措施
Historic behaviour vs. physical measures for irrational flows with multiple stopping points
论文作者
论文摘要
我们研究了伯克霍夫（Birkhoff）的平均值，沿两个停止点（例如$ \ mathbf p $ and $ \ mathbf q $）的两个停止点的非理性线性流的平稳重新聚集轨迹。这种平均值的限制行为与一组完整的Haar-Lebesgue度量中的起点无关，并且以复杂的方式取决于线性流的斜率$α$的Diophantine属性，以及$ \ MATHBF P $和$ \ MATHBF Q $的相对位置。尤其是，如果$α$是Diophantine，那么Birkhoff的限制几乎到处都有差异（历史行为），如果$α$足够liouville，那么就存在一些$ \ Mathbf P $和$ \ Mathbf Q $，因此Birkhoff平均几乎可以融合到任何地方（独特的物理测量）。
We study Birkhoff averages along trajectories of smooth reparameterizations of irrational linear flows of the two torus with two stopping points, say $\mathbf p$ and $\mathbf q$, of quadratic order. The limiting behaviour of such averages is independent of the starting point in a set of full Haar-Lebesgue measure and depends in an intricate way on the Diophantine properties of both the slope $α$ of the linear flow as well as the relative position of $\mathbf p$ and $\mathbf q$. In particular, if $α$ is Diophantine, then Birkhoff limits diverge almost everywhere (historic behaviour) and if $α$ is sufficiently Liouville, then there exists some $\mathbf p$ and $\mathbf q$ such that the Birkhoff averages converge almost everywhere (unique physical measure).
