诱导位移映射的动力学
Dynamics of the Induced Shift Map
摘要：
在这篇文章中，我们比较了移位空间的超空间上的移位映射及其诱导映射的动力学。我们证明了诱导移位映射的许多性质可以很容易地通过适当的符号序列来证明。我们比较了移位系统$(Ω, σ)$与它的诱导对应物$(\mathcal{K}(Ω),\overlineσ)$的动力学，其中$\mathcal{K}(Ω)$是$Ω$的所有非空紧子集的超空间。最近，这种比较已经研究了很多一般空间。我们继续同样的研究的情况下，转移空间，并提出了这样的研究序列的意义。我们比较了移位映射及其导出映射的混合性质、周期点的稠密性、各种形式的灵敏性和可扩张性。特别是，我们证明了它们的等价性的情况下，充分的转变。我们还研究了有限型子移位的特殊情况，特别证明了弱混合、混合和灵敏性在两个系统中是等价的。对于任何一般子移位，我们证明了上有限灵敏度的概念在两个系统中是等价的，并且对于传递子移位，上有限灵敏度等价于共分灵敏度。在这个过程中，我们证明了所有的sturmian子移位是余弦敏感的。
英文摘要：
In this article, we compare the dynamics of the shift map and its induced counterpart on the hyperspace of the shift space. We show that many of the properties of induced shift map can be easily demonstrated by appropriate sequences of symbols. We compare the dynamics of the shift system $(Ω, σ)$ with its induced counterpart $(\mathcal{K}(Ω),\overlineσ)$, where $\mathcal{K}(Ω)$ is the hyperspace of all nonempty compact subsets of $Ω$. Recently, such comparisons have been studied a lot for general spaces. We continue the same study in case of shift spaces, and bring out the significance of such a study in terms of sequences. We compare the mixing properties, denseness of periodic points, various forms of sensitivity and expansivity of the shift map and its induced counterpart. In particular, we show their equivalence in case of the full shift. We also look into the special case of subshifts of finite type, and in particular prove that the properties of weakly mixing, mixing and sensitivity are equivalent in both systems. And in the case of any general subshift, we show that the concept of cofinite sensitivity is equivalent in both systems and for transitive subshifts, cofinite sensitivity is equivalent to syndectic sensitivity. In the process we prove that all sturmian subshifts are cofinitely sensitive.
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