An area-preserving diffeomorphism of an annulus has an “action function” which measures how the diffeomorphism distorts curves. The average value of the action function over the annulus is known as the Calabi invariant of the diffeomorphism, while the average value of the action function over a periodic orbit of the diffeomorphism is the mean action of the orbit. If an area-preserving annulus diffeomorphism is a rotation near the boundary, and if its Calabi invariant is less than the maximum boundary value of the action function, then we show that the infimum of the mean action over all periodic orbits of the diffeomorphism is less than or equal to its Calabi invariant.

中文翻译：

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保面积环微分同胚周期轨道的平均作用

环的面积保持微分同胚具有“作用函数”，它测量微分同胚如何扭曲曲线。作用函数在环面上的平均值称为微分同胚的 Calabi 不变量，而作用函数在微分同胚周期轨道上的平均值称为轨道的平均作用。如果保面积环微分同胚是边界附近的旋转，并且如果它的卡拉比不变量小于作用函数的最大边界值，那么我们证明了微分同胚所有周期轨道上的平均作用的下确界小于大于或等于其 Calabi 不变量。