Dynamical properties of shift maps on inverse limits with a set valued function
Dynamical properties of shift maps on inverse limits with a set valued function
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具有集值函数的反极限位移映射的动态特性
DOI：
10.1017/etds.2016.73
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发表时间：
2016
影响因子：
0.9
通讯作者：
V. Nall
中科院分区：
文献类型：
--
作者：
J. Kennedy;V. Nall
从区间到区间的闭子集的集值函数出现在科学和数学建模的各个领域。研究表明，紧致空间上的单值函数的动力学与以该函数为唯一键映射的逆极限上的移位映射的动力学密切相关。例如，它已被证明，与Devaney的混沌定义的键函数是混沌的，当且仅当移位映射是混沌的。关注这种联系的一个原因是移位映射是逆极限上的同胚，因此逆极限空间的拓扑结构必须反映移位映射的动力学。在集值情况下，可能没有混沌的自然定义，因为每个点都没有一个明确定义的轨道。然而，移位映射是一个连续的单值函数，因此它与逆极限空间一起形成了一个动力系统，该系统在任何通常意义下都可以是混沌的。对于集值情形，我们用定理和例子证明了当移位映射是混沌的（在某些不变集上）时，逆极限中丰富的拓扑结构。然后，我们连接到一个属性的集值函数，这是一个重要的连续函数的混沌产生属性的自然推广的混乱。
Set-valued functions from an interval into the closed subsets of an interval arise in various areas of science and mathematical modeling. Research has shown that the dynamics of a single-valued function on a compact space are closely linked to the dynamics of the shift map on the inverse limit with the function as the sole bonding map. For example, it has been shown that with Devaney’s definition of chaos the bonding function is chaotic if and only if the shift map is chaotic. One reason for caring about this connection is that the shift map is a homeomorphism on the inverse limit, and therefore the topological structure of the inverse-limit space must reflect in its richness the dynamics of the shift map. In the set-valued case there may not be a natural definition for chaos since there is not a single well-defined orbit for each point. However, the shift map is a continuous single-valued function so it together with the inverse-limit space form a dynamical system which can be chaotic in any of the usual senses. For the set-valued case we demonstrate with theorems and examples rich topological structure in the inverse limit when the shift map is chaotic (on certain invariant sets). We then connect that chaos to a property of the set-valued function that is a natural generalization of an important chaos producing property of continuous functions.