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COMPUTABILITY OF TOPOLOGICAL ENTROPY: FROM GENERAL SYSTEMS TO TRANSFORMATIONS ON CANTOR SETS AND THE INTERVAL

机译:拓扑熵的可计算性:从一般系统到唱名集的变换和间隔

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The dynamics of symbolic systems, such as multidimensional sub-shifts 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 algorithmic. 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}~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}~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.
机译:已知符号系统的动态,例如有限型或蜂窝自动机的多维子变换,与可计算性理论密切相关。特别是,用于描述和对这种系统的拓扑熵描述和分类拓扑熵的适当工具是算法。这些符号系统的一部分非常重要依赖于他们在了解更多常规系统中的作用,而不是非符号空间。本文的目的是从更一般的,不一定符号设置来调查从可计算性的观点来看拓扑熵。类似于有效的子筛选,我们考虑通过一般度量空间中的有效紧凑型集的可计算映射,并研究其拓扑熵的可计算性属性。我们展示即使在这个常规设置中,熵始终是Σ_2可计算的数字。然后,我们研究各种动态和分析约束如何影响这种上限,并证明它可以根据所考虑的约束以不同的方式降低。特别地,我们可以在{0,1}〜n上的设置映射映射的类别中获得所有Σ_2可计算的数字,但是当限于膨胀地图时,这种绑定减少到π_1(或上部)履调的数字。另一方面,如果我们将环境空间的几何形状从符号{0,1}〜n更改为单位间隔[0,1],那么我们发现了一个完全不同的情况 - 我们显示可计算的可能的熵超过[0,1]的系统正好是Σ_1(或更低)的追容数字,并且当我们将系统类别限制为二次族时,该表征将关闭以精确地切换到可计算的数字。

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