arXiv 1906.017452019-06-04math.DScs.CC

关于拓扑熵可计算性的一般方法

On the computability properties of topological entropy: a general approach

Silvere Gangloff, Alonso Herrera, Cristobal Rojas, Mathieu Sablik

备注：: 27 pages, 5 figures

摘要：

符号系统的动力学，如有限型或元胞自动机的多维子移位，被认为与可计算性理论密切相关。特别是，适当的工具来描述和分类的拓扑熵这类系统原来是计算的性质。这些符号系统的重要性部分取决于它们在理解非符号空间上的更一般系统方面所起的作用。本文的目的是调查拓扑熵从可计算性的角度来看，在这个更一般的，不一定是符号设置。类似于有效子移位，我们考虑了一般度量空间中有效紧集上的可计算映射，并研究了它们的拓扑熵的可计算性。我们表明，即使在这个一般的设置，熵总是一个$Σ_2$ -可计算的数字。然后，我们研究了各种动力学和分析约束如何影响这个上界，并证明了它可以降低以不同的方式取决于所考虑的约束。特别是，我们得到的所有$Σ_2$ -可计算数已经可以实现满射可计算映射类内的$\{0,1\}^{\mathbb{N}}$，但这一界减少到$Π_{1}$（或上）-可计算数时，限制到扩张的地图。另一方面，如果我们将周围空间的几何形状从符号$\{0,1\}^{\mathbb{N}}$改变为单位间隔$[0,1]$，然后我们发现了一个完全不同的情况--我们证明了在$[0,1]$上可计算系统的可能熵恰好是$Σ_{1}$（或更低）-可计算的数字和这个特征切换到精确的可计算的数字时，我们限制系统的类的二次家庭。

英文摘要：

The dynamics of symbolic systems, such as multidimensional subshifts of finite type or cellular automata, are known to be closely related to computability theory. In particular, the appropriate tools to describe and classify topological entropy for this kind of systems turned out to be of computational nature. Part of the great importance of these symbolic systems relies on the role they have played in understanding more general systems over non-symbolic spaces. The aim of this article is to investigate topological entropy from a computability point of view in this more general, not necessarily symbolic setting. In analogy to effective subshifts, we consider computable maps over effective compact sets in general metric spaces, and study the computability properties of their topological entropies. We show that even in this general setting, the entropy is always a $Σ_2$-computable number. We then study how various dynamical and analytical constrains affect this upper bound, and prove that it can be lowered in different ways depending on the constraint considered. In particular, we obtain that all $Σ_2$-computable numbers can already be realized within the class of surjective computable maps over $\{0,1\}^{\mathbb{N}}$, but that this bound decreases to $Π_{1}$(or upper)-computable numbers when restricted to expansive maps. On the other hand, if we change the geometry of the ambient space from the symbolic $\{0,1\}^{\mathbb{N}}$ to the unit interval $[0,1]$, then we find a quite different situation -- we show that the possible entropies of computable systems over $[0,1]$ are exactly the $Σ_{1}$(or lower)-computable numbers and that this characterization switches down to precisely the computable numbers when we restrict the class of system to the quadratic family.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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