Identifying invariant ergodic subsets and barriers to mixing by cutting and shuffling: Study in a birotated hemisphere.,Physical Review E

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Identifying invariant ergodic subsets and barriers to mixing by cutting and shuffling: Study in a birotated hemisphere.
Physical Review E ( IF 2.2 ) Pub Date : 2020-01-01 , DOI: 10.1103/physreve.101.012204
Thomas F Lynn ₁ , Julio M Ottino ₂ , Paul B Umbanhowar ₃ , Richard M Lueptow ₄

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Mixing by cutting and shuffling can be mathematically described by the dynamics of piecewise isometries (PWIs), higher dimensional analogs of one-dimensional interval exchange transformations. In a two-dimensional domain under a PWI, the exceptional set, E[over ¯], which is created by the accumulation of cutting lines (the union of all iterates of cutting lines and all points that pass arbitrarily close to a cutting line), defines where mixing is possible but not guaranteed. There is structure within E[over ¯] that directly influences the mixing potential of the PWI. Here we provide computational and analytical formalisms for examining this structure by way of measuring the density and connectivity of ɛ-fattened cutting lines that form an approximation of E[over ¯]. For the example of a PWI on a hemispherical shell studied here, this approach reveals the subtle mixing behaviors and barriers to mixing formed by invariant ergodic subsets (confined orbits) within the fractal structure of the exceptional set. Some PWIs on the shell have provably nonergodic exceptional sets, which prevent mixing, while others have potentially ergodic exceptional sets where mixing is possible since ergodic exceptional sets have uniform cutting line density. For these latter exceptional sets, we show the connectivity of orbits in the PWI map through direct examination of orbit position and shape and through a two-dimensional return plot to explain the necessity of orbit connectivity for mixing.

中文翻译：

通过切割和改组来确定遍历遍历的子集和混合障碍：在双旋半球中进行研究。

剪切和混洗的混合可以通过分段等距（PWI）的动力学数学描述，即一维间隔交换变换的高维类似物。在PWI下的二维域中，例外集E [over]由切割线的累积（切割线的所有迭代和任意靠近切割线的所有点的并集）创建。定义在哪里可以混合但不能保证混合。E [over]内有一种结构直接影响PWI的混合电位。在这里，我们提供了计算和分析形式上的方法，以通过测量形成E [over]近似值的att型切割线的密度和连通性来检查这种结构。以此处研究的半球形壳上的PWI为例，这种方法揭示了特殊集合的分形结构内不变的遍历子集（受限轨道）形成的细微混合行为和混合障碍。外壳上的某些PWI具有可证明是非遍历的特殊设置，可防止混合，而其他PWI具有可能遍历的可能遍历的特殊设置，因为遍历的特殊设置具有统一的切割线密度。对于后面的这些例外集合，我们通过直接检查轨道的位置和形状以及通过二维返回图来说明PWI图中轨道的连通性，以解释混合时轨道连通性的必要性。外壳上的某些PWI具有可证明是非遍历的特殊设置，可防止混合，而其他PWI具有可能遍历的可能遍历的特殊设置，因为遍历的特殊设置具有统一的切割线密度。对于后面的这些例外集合，我们通过直接检查轨道的位置和形状以及通过二维返回图来说明PWI图中轨道的连通性，以解释混合时轨道连通性的必要性。外壳上的某些PWI具有可证明是非遍历的特殊设置，可防止混合，而其他PWI具有可能遍历的可能遍历的特殊设置，因为遍历的特殊设置具有统一的切割线密度。对于后面的这些例外集合，我们通过直接检查轨道的位置和形状以及通过二维返回图来说明PWI图中的轨道连通性，以解释混合时轨道连通性的必要性。

更新日期：2020-01-10

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