[论文解读] Cantor proximal systems with topological rank 2 are residually scrambled

Takashi Shimomura|arXiv (Cornell University)|Feb 4, 2016

Mathematical Dynamics and Fractals被引用 2

一句话总结

本文研究了一类零维、本质上极小的动力系统，其拓扑秩为2，且唯一极小集由一个不动点构成。通过布拉特利图和拓扑秩理论，证明了所有此类渐近等价的康托尔系统均为剩余混沌，建立了唯一遍历性的条件，并刻画了遍历测度的数量——表明其可为1或2，具体取决于混合性质，且给出了弱混合、拓扑混合及唯一遍历性情形下的例子。

ABSTRACT

Downarowicz and Maass (2008) proposed topological ranks for all homeomorphic Cantor minimal dynamical systems using properly ordered Bratteli diagrams. In this study, we adopt this definition to the case of all essentially minimal zero-dimensional systems. We consider the cases in which topological ranks are 2 and unique minimal sets are fixed points. Akin and Kolyada (2003), in their study of Li--Yorke sensitivity, showed that if the unique minimal set of an essentially minimal system is a fixed point, then the system must be proximal. However, a finite topological rank implies expansiveness; furthermore, in the case of proximal Cantor systems with topological rank 2, the expansiveness is always from the lowest degree. Rank 2 zero-dimensional systems might be thought as a part of the rank 1 transformations that are considered in the vast field of ergodic theory. However, these systems are also interesting from the perspective of topological chaos theory; e.g., in this study, we show that all proximal Cantor systems with topological rank 2 are residually scrambled. In addition, we investigate the finite invariant measures on these systems. Evidently, such systems have at most two ergodic measures. We present a necessary and sufficient condition for the unique ergodicity of these systems. In addition, we show that the number of ergodic measures of systems that are topologically mixing can be 1 and 2. Moreover, we present examples that are topologically weakly mixing, not topologically mixing, and uniquely ergodic. Finally, we show that the number of ergodic measures of systems that are not weakly mixing can be 1 and 2.

研究动机与目标

• 将拓扑秩理论扩展至具有唯一不动点极小集的本质极小零维系统。
• 确定拓扑秩为2的渐近等价康托尔系统的动力行为，特别是混沌性与遍历性。
• 基于混合性质，对这类系统中遍历测度的数量进行分类。
• 构造展示拓扑弱混合、非混合及唯一遍历性的例子。

提出的方法

• 采用通过适当排序的布拉特利图定义拓扑秩，以适用于零维系统。
• 分析具有拓扑秩2和不动点极小集的渐近等价康托尔系统的结构。
• 应用Akin与Kolyada（2003）关于极小系统中Li–Yorke敏感性与渐近等价性的结果。
• 利用有限拓扑秩导出的扩张性质来刻画系统动力学。
• 为这些系统建立唯一遍历性的必要与充分条件。
• 构造显式例子，以展示弱混合、非混合及唯一遍历行为。

实验结果

研究问题

• RQ1 所有拓扑秩为2的渐近等价康托尔系统是否均为剩余混沌？
• RQ2 在具有不动点极小集的拓扑秩为2的零维系统中，什么条件可保证唯一遍历性？
• RQ3 拓扑混合如何影响此类系统中遍历测度的数量？
• RQ4 此类系统是否可以是弱混合、拓扑混合或唯一遍历的？各类情形的特征是什么？
• RQ5 在非弱混合系统中，可能存在的遍历测度的最大数量是多少？

主要发现

• 所有拓扑秩为2的渐近等价康托尔系统均为剩余混沌，确认了强形式的拓扑混沌。
• 此类系统最多具有两个遍历测度，其数量取决于混合性质。
• 为这些系统建立了唯一遍历性的必要与充分条件。
• 拓扑混合系统可能具有一个或两个遍历测度。
• 存在展示拓扑弱混合、非拓扑混合及唯一遍历性的例子。
• 对于非弱混合系统，其遍历测度的数量可能为一个或两个。