零拓扑熵的词的代价和维数
Cost and dimension of words of zero topological entropy
摘要：
设$A^*$表示由有限非空集$A.$生成的自由幺半群。本文引入了一种新的语言复杂性度量$L\subseteq A^*$，它是根据$A^*.$上的半群结构定义的。对于每一个$L\subseteq A^*,$，我们定义了它的{ \it代价} $c(L)$作为所有实数$α$的下确界，对于这些实数，存在具有$p_S(n)=O(n^α)$的语言$S\subseteq A^*$和具有$L\subseteq S^k.$的正整数$k$。{ \it成本维数} $d_c(L)$作为所有正整数$k$的集合的下确界，使得$L\subseteq S^k$对于某些语言$S$具有$p_S(n)=O(n^{c(L)}).$我们主要对由零拓扑熵的无限字$x=x_0x_1x_2\cdots \in A^ω$的因子集合给出的语言$L$感兴趣，在这种情况下$c(L)<+\infty.$我们建立线性因子复杂度的字的以下表征：令$x\in A^ω$和$L=$ FacPDMATHABNH是$x.$的因子集，则$p_x(n)=Θ(n)$当且仅当$c(L)=0$和$d_c(L)=2.$换句话说，$p_x(n)=O(n)$当且仅当Fac $(x)\subseteq S^2$对于某些有界复杂度的语言$S\subseteq A^+$（含义$\limsup p_S(n)<+\infty).$一般来说，语言$L$的代价深刻地反映了由半群结构引起的底层组合结构，例如，与上述由次线性复杂度的词生成的语言的特征相反，存在复杂度为$p_L(n)=O(\log n)$的非阶乘语言$L$（因此成本等于$0)$，成本维数为$+\infty.$）。在本文中，我们研究了由零拓扑熵的无限词定义的语言的成本和成本维数。
英文摘要：
Let $A^*$ denote the free monoid generated by a finite nonempty set $A.$ In this paper we introduce a new measure of complexity of languages $L\subseteq A^*$ defined in terms of the semigroup structure on $A^*.$ For each $L\subseteq A^*,$ we define its {\it cost} $c(L)$ as the infimum of all real numbers $α$ for which there exist a language $S\subseteq A^*$ with $p_S(n)=O(n^α)$ and a positive integer $k$ with $L\subseteq S^k.$ We also define the {\it cost dimension} $d_c(L)$ as the infimum of the set of all positive integers $k$ such that $L\subseteq S^k$ for some language $S$ with $p_S(n)=O(n^{c(L)}).$ We are primarily interested in languages $L$ given by the set of factors of an infinite word $x=x_0x_1x_2\cdots \in A^ω$ of zero topological entropy, in which case $c(L)<+\infty.$ We establish the following characterisation of words of linear factor complexity: Let $x\in A^ω$ and $L=$Fac$(x)$ be the set of factors of $x.$ Then $p_x(n)=Θ(n)$ if and only $c(L)=0$ and $d_c(L)=2.$ In other words, $p_x(n)=O(n)$ if and only if Fac$(x)\subseteq S^2$ for some language $S\subseteq A^+$ of bounded complexity (meaning $\limsup p_S(n)<+\infty).$ In general the cost of a language $L$ reflects deeply the underlying combinatorial structure induced by the semigroup structure on $A^*.$ For example, in contrast to the above characterisation of languages generated by words of sub-linear complexity, there exist non factorial languages $L$ of complexity $p_L(n)=O(\log n)$ (and hence of cost equal to $0)$ and of cost dimension $+\infty.$ In this paper we investigate the cost and cost dimension of languages defined by infinite words of zero topological entropy.
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