Lynn Thomas F, Ottino Julio M, Umbanhowar Paul B, Lueptow Richard M
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA.
Department of Chemical and Biological Engineering, Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208, USA and Northwestern Institute on Complex Systems (NICO), Northwestern University, Evanston, Illinois 60208, USA.
Phys Rev E. 2020 Jan;101(1-1):012204. doi: 10.1103/PhysRevE.101.012204.
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.
通过切割和洗牌进行的混合可以用分段等距变换（PWIs）的动力学进行数学描述，它是一维区间交换变换的高维类似物。在二维域中，在分段等距变换下，由切割线的累积（切割线的所有迭代以及任意接近切割线的所有点的并集）产生的例外集(\overline{E})，定义了混合可能发生但不保证发生的位置。(\overline{E})内部存在直接影响分段等距变换混合潜力的结构。在这里，我们提供了计算和分析形式体系，通过测量形成(\overline{E})近似的(\epsilon)-加厚切割线的密度和连通性来研究这种结构。对于此处研究的半球壳上的分段等距变换示例，这种方法揭示了由例外集分形结构内的不变遍历子集（受限轨道）形成的微妙混合行为和混合障碍。壳上的一些分段等距变换具有可证明的非遍历例外集，这会阻止混合，而其他一些则具有潜在遍历的例外集，在这些例外集中混合是可能的，因为遍历例外集具有均匀的切割线密度。对于后一种例外集，我们通过直接检查轨道位置和形状以及通过二维返回图来展示分段等距变换映射中轨道的连通性，以解释轨道连通性对于混合的必要性。
