多维子移位中扩展集的可计算性：渐近增长、动力学约束
Computability of extender sets in multidimensional subshifts: asymptotic growths, dynamical constraints
AI总结：
本研究针对多维子移位扩展集的可计算性展开，通过扩展熵与扩展熵维度分析其增长特性，证明sofic与有效子移位的扩展熵及维度取值范围一致，并研究了不同约束下这类增长指标的计算复杂度。
摘要：
子移位是由一族禁用模式定义的$\n\mathbb{Z}^d$上的着色集合。在给定子移位中，有限模式的扩展集是其所有可容许补全的集合。由于$\mathbb{Z}$子移位的sofic性等价于扩展集数量有限，学界曾推测扩展集的数量可作为区分高维sofic子移位与有效子移位类别的依据。我们研究了多维子移位中扩展集的若干计算特性，重点关注其增长性，指标包括扩展熵（arXiv:1711.07515）和扩展熵维度。本文证明sofic子移位与有效子移位具有相同的可能扩展熵（恰好是$[0,+\infty)$中的$Π_3$-可计算实数）和扩展熵维度，并探究了这些增长类量在不同动力学与组合约束下的计算复杂度。
英文摘要：
Subshifts are sets of colorings of $\mathbb{Z}^d$ defined by families of forbidden patterns. In a given subshift, the extender set of a finite pattern is the set of all its admissible completions. Since soficity of $\mathbb{Z}$ subshifts is equivalent to having a finite number of extender sets, it had been conjectured that the number of extender sets could provide a way to separate the classes of sofic and effective subshifts in higher dimensions. We investigate some computational characterizations of extender sets in multidimensional subshifts, and in particular their growth, in terms of extender entropies (arXiv:1711.07515) and extender entropy dimensions. We prove here that sofic and effective subshifts have the same possible extender entropies (exactly the $Π_3$-computable real numbers of $[0,+\infty)$) and extender entropy dimensions, and investigate the computational complexity of these growth-type quantities under various dynamical and combinatorial constraints.
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